ft— h_Ji_ 


REESE  LIBRARY 

01    i in: 

UNIVERSITY  OF  CALIFORNIA 


n — n — n, 


THJE 

PRINCIPLES 

OF 


MATHEMATICAL  CHEMISTRY. 

THE  ENERGETICS   OF  CHEMICAL 
PHENOMENA 


BY 


DR.   GEORG    HELM, 

Professor  in  the  Royal  Technical  High  School,  Dresden, 


AUTHORIZED   TRANSLATION  FROM   THE  GERMAN 

BY 

J.  LIVINGSTON    R.  MORGAN,  PH.D.  (LEIPZIG,) 

Instructor  in  the  Brooklyn  Polytechnic  Institute, 


FIRST   EDITION 
FIRST    THOUSAND. 


NEW   YORK: 

JOHN     WILEY    &     SONS. 

LONDON  :   CHAPMAN  &   HALL,   LIMITED. 

1897. 


Copyright,  1897, 

BY 

J.  L.  R.  MORGAN. 


ROBERT  DRUMMOND,   ELECTROTYPER   AND   PRINTER,   NEW  YORK. 


TRANSLATOR'S  PREFACE. 


IN  presenting  this  work  to  the  English-speaking 
public,  the  translator  is  actuated  by  the  desire  to 
spread  the  knowledge  o.L  Physical  Chemistry  more 
widely  among  students  and  chemists.  The  knowledge 
gained  by  it  is  of  undoubted  value,  in  all  parts  of 
chemical  science,  and  is  gaining  in  importance  daily ; 
and  the  time  is  coming  when  no  chemist,  whatever  his 
branch,  will  .be  considered  completely  equipped  with- 
out it.  In  translating,  the  text  has  been  followed  as 
closely  as  possible,  and  clearness  rather  than  literary 
style  has  been  the  aim.  Dr.  Helm  has  been  so  good 
as  to  read  the  work,  and  the  thanks  of  the  translator 
can  best  be  expressed  to  him  in  this  place. 

iii 


PREFACE  TO  THE  GERMAN  EDITION. 


THE  title  "  Mathematical  Chemistry  "  means  that 
the  purpose  of  this  little  book  is  to  collect  the  results, 
according  to  the  deductive  method,  of  the  investiga- 
tions in  the  realm  of  general  chemistry. 

The  subject  of  the  mathematical  consideration  of 
nature,  which  in  its  earlier  development  was  known  as 
Physical  Chemistry,  can  now,  in  its  present  state,  be 
viewed  from  a  general  theoretical  standpoint  as  a 
whole ;  and  in  this  state  it  appears  as  one  of  ~the  clear- 
est and  most  complete  proofs  of  the  principle  of  the 
conservation  of  energy.  The  fact  that  single  parts  off 
mathematical  chemistry  are  derived  from  other  reason- 
ing, without  regard  to,  or  with  mere  passing  considera- 
tion of,  this  general  principle— as,  e.g.,  the  conception 
of  the  Osmotic  Pressure  from  the  analogy  to  gases,  or 
from  hypothetical  molecular  theories — can  be  attrib- 
uted to  the  difference  in  the  points  of  departure  of 
these  investigations,  and  also  to  the  earlier  lack  of  rec- 
ognition of  the  Theory  of  Gibbs. 

x  The  return  to  Willard  Gibbs  and — as  far  as  con- 
cerns thermodynamics  proper — to  Horstmann  appears 
to  me  to  be  a  purification  of  the  scientific  structure 


VI  PREFACE    TO    THE   GERMAN  EDITION. 

from  conceptions  which  have  become  unnecessary  to 
it.  In  the  light  of  fewer  conceptions — held  together 
by  the  principle  of  energy — we  have  a  complete  view 
of  the  whole,  which  is  particularly  desirable  in  a  first 
(introduction  to  the  subject. 

The  presence  of  experiments  in  this  book  could 
hardly  be  allowed  ;  for  they  do  not  here,  as  in  the  case 
of  the  inductive  presentation,  lead  to  the  formation  of 
the  ideas,  but  only  serve  to  elucidate  them  and  to 
make  the  reader  familiar  with  their  application.  On 
the  other  hand,  Ostwald's  very  complete  book  relieves 
the  author  from  dwelling  upon  the  details  of  the  single 
investigations.  If  I,  according  perhaps  to  the  taste  of 
some  of  my  readers,  have  restricted  myself  too  much 
in  this  respect,  it  is  with  the  hope  that,  by  it  the  lead- 
ing conception,  in  each  case,  may  be  the  more  promi- 
nent, and  impress  itself  upon  the  student  by  the  sim- 
plicity of  its  application. 

GEORG  HELM. 

DRESDEN,  Aug.  1894. 


CONTENTS. 


PART  I. 

ENERGY. 

CHAPTER  PAGE 

I.  THE  APPLICATION  OF  THE  PRINCIPLE  OF  THE  CONSER- 
VATION OF  ENERGY  TO  CHEMICAL  REACTIONS i 

II.  THE  MEASUREMENT  OF  CHEMICAL  ENERGY 4 

III.  CHANGES  IN  TEMPERATURE  AND  IN  THE  STATE  OF  AGGRE- 

GATION   15 

IV.  MECHANICAL  ENERGY 19 

V.  THE  VOLUME  ENERGY  OF  GASES 25 

PART  II. 

ENTROPY. 

I.  THE  FACTORS  OF  ENERGY 41 

II.  THE  THERMODYNAMICS  OF  PERFECT  GASES 48 

III.  THE  CYCLE 56 

IV.  THE  ENTROPY  OF  GASES  AND  GAS  MIXTURES 67 

V.  THE  RELATIONS  BETWEEN  HEAT  AND  VOLUME  ENERGY.  .  72 

a.  Isothermal  Changes  in  a  System  of  Two  Phases  of 

the  Same  Substance 79 

b.  Changes  in  the  State  of  Aggregation 82 

c.  Allotropic  Changes 88 

i.   Dissociation 89 

e.  The  Heat  of  Solution  and  Dilution 94 

VI.  THE  RELATION  BETWEEN  HEAT  AND  ELECTRICAL  ENERGY  96 

vii 


Vlll  CONTENTS. 

PART  III. 

THE  CHEMICAL  INTENSITY. 

CHAPTER  PAGE 

I.  THE  GENERAL  PROPERTIES  OF  CHEMICAL  INTENSITY..  . ,.  114 
II.  THE  SIMPLE  CHEMICAL  REACTION 130 

III.  CHEMICAL  EQUILIBRIUM 141 

IV.  FREEZING  AND  BOILING  POINTS,  ALSO  VAPOR  PRESSURES 

OF  HIGHLY  DILUTED  SOLUTIONS 161 

V.  OSMOTIC  PRESSURE 174 

VI.  DIFFUSION 182 

VII.  THE  VELOCITY  OF  A  CHEMICAL  REACTION 192 

PART  IV. 

THE  DEGREES  OF  FREEDOM  OF  CHEMICAL  PHENOMENA. 

I.  THE  RULE  OF  PHASES 202 

II.  THE  EQUILIBRIUM  OF  PHASES 209 

III.  CHEMICAL  REACTIONS  THAT  DEPEND  UPON  SEVERAL  PA- 
RAMETERS   214 


THE  PRINCIPLES  OF 
MATHEMATICAL  CHEMISTRY. 


PART  I. 
ENERGY. 


CHAPTER  I. 

THE  APPLICATION   OF  THE    PRINCIPLE    OF  THE    CON- 
SERVATION  OF  ENERGY   TO   CHEMICAL   REACTIONS. 

THE  basis  of  the  mathematical  treatment  of  chemi- 
cal phenomena  is  the  principle  of  the  conservation  of 
energy, — which  has  shown  itself  in  no  branch  of  the 
mathematical  knowledge  of  nature  so  valuable  or  suffi- 
cient as  in  those  two  nearly  related  processes,  viz., 
thermal  and  chemical  reactions. 

The  principle  of  the  conservation  of  energy  is  best 
used  for  the  study  of  chemical  phenomena  in  the 
following  form  :* 

The  momentary  state  of  a  body  or  part  of  a  body 
is  determined  by  certain  varying  quantities,  as  co- 

*  For  the  gradual  development  of  this  treatment  of  natural  phe- 
nomena see  the  author's  Lehre  von  der  Energie  (Leipzig,  1887), 


2      PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 

ordinates  in  space,  velocity,  temperature,  electrical 
charge,  etc.,  which  are  called  parameters  of  that  body. 
'Then  for  each  of  the  smallest  particles  of  the  body,  in 
each  instant,  we  have  a  certain  varying  quantity,  its 
intrinsic  energy,  *  which  depends  only  upon  the  mo- 
mentary state  of  the  body,  and  which  is  therefore  a 
function  of  a  parameter.  This  function  has  the  prop- 
erty by  which  every  possible  change  in  Nature  can  be 
considered  as  an  increase  of  the  intrinsic  energy  of 
some  bodies  and  a  decrease  of  the  same  in  others,  and 
by  which  the  total  amount  of  all  intrinsic  energies,  by 
?all  changes  in  Nature,  must  remain  unchanged. 

The  principle  of  the  conservation  of  energy  so  ex- 
pressed is  not  to  be  easily  proven,  but  it  presents  to  us 
a  way  of  considering  natural  phenomena,  the  correct- 
ness of  which  can  only  be  established  by  the  results. 
Thus  far,  in  very  many  cases,  it  has  been  possible 
to  define  the  intrinsic  energy  of  a  body  undergoing 
change  in  such  a  manner  that  this  method  has  been 
shown  to  be  possible,  and  irfdeed  to  be  preferred  ;  and 
experiments  have  always  shown  that  the  difference 
between  two  intrinsic  energies,  so  defined,  is  independ- 
ent of  the  means  by  which  the  one  is  transformed  into 
the  other. 

As  far  as  concerns  chemical  changes,  Lavoisier  and 

*  The  intrinsic  energy  of  a  body  in  a  given  state  represents  the 
mechanical  value  of  all  actions  which  the  body  would  exert  by  trans- 
formation from  a  given  state  into  the  standard  state,  or  the  mechani- 
cal value  of  all  efforts  which  are  necessary  to  take  the  body  from  the 
standard  state  into  the  given  state.  (Wm.  Thomson,  Math.  Phys. 
Papers,  I,  292.) — TRANS. 


THE   CONSER  VA  TION  OF  EN  ERG  Y.  3 

Laplace,  in  1780,  considered  it  self-evident  that  all 
increase  and  decrease  of  heat  which  is  shown  by  a 
change  of  state  in  a  system  of  bodies  is  also  shown  in 
the  reverse  order  when  the  system  returns  to  its  initial 
state.  To  be  sure,  this  view  was  adopted  from  the 
experience  of  such  reversible  changes  as  solution  and 
changes  in  the  state  of  aggregation,  by  the  reversal  of 
which  all  the  intermediate  states  follow  in  the  reverse 
order.  But  it  is  just  in  the  case  of  pure  chemical  reac- 
tions that  this  reversibility  rarely  occurs.  Hess,*  how- 
ever, in  1840,  proved  that,  in  general,  two  chemical 
reactions  that  transform  the  state  A  into  the  state  B 
develop  the  same  amount  of  energy;  i.e.;  the  differ- 
ence in  energy  between  the  states  A  and  B  depends 
only  on  these  states,  and  not  upon  the  nature  of  the 
transformation.  He  transformed  HQSO4  by  addition 
of  water  into  a  certain  hydrate  of  sulphuric  acid,  and 
this  by  addition  of  ammonia  into  a  water  solution  of 
ammonium  sulphate.  He  found  by  this  that  when 
the  reaction  took  place  in  a  calorimeter  the  same 
amount  of  heat  was  developed,  within  the  limits  of 
experimental  error,  independent  of  which  hydrate 
had  been  formed  first ;  and  this  amount  was  the  same 
as  that  developed  by  the  immediate  formation  of 
(NH4)2SO4  +  Aq  by  the  action  of  ammonia  and  pure 
sulphuric  acid,  these  experiments  have  since,  by 
more  accurate  methods  of  observation,  been  confirmed. 

*  Pogg.  Ann.  50.     Ostwald's  Klassiker  d.  exakt.  Wiss.,  vol.  ix. 


CHAPTER  II. 

THE  MEASUREMENT  OF  CHEMICAL  ENERGY. 

SINCE  in  all  natural  phenomena  the  intrinsic  en- 
ergies are  only  noticeable  by  their  changes,  the  only 
possible  way  we  have  o.f  measuring  them  is  by  their 
differences.  As  we  cannot  deprive  a  body  of  all  its 
energy,  it  is  impossible  to  measure  all  that  is  in  it,  and 
consequently  we  must  be  satisfied  with  the  knowledge 
of  how  much  it  exceeds  or  falls  short  of  the  energy  of 
that  body  in  a  certain  definite  state  which  we  call  the 
•  standard  state  (Normalzustand). 

The  simplest  way  of  measuring  the  difference  be- 
tween the  intrinsic  energy  of  a  body  in  the  state  A, 
e.g.,  the  standard  state,  and  another  state,  B,  is  to 
transform  the  one  into  the  other  in  such  a  manner 
that  the  difference  of  energy  appears  in  one  form, — as 
heat,  in  a  calorimeter.  In  this  way,  in  many  cases,  we 
can  measure  directly  the  difference  of  energy  in  the 
chemical  reaction  by  the  heat  generated. 

As  unit  of  heat  or  calorie  (cal.),  it  is  best  to  choose, 
in  accord  with  the  other  units  employed,  the  heat 
energy  which  is  necessary  to  warm  the  mass  of  one 
gram  of  water  i°  Celsius  (air-thermometer). 

Since  the  specific  heat  of  water  is  a  function  of  the 
temperature,  it  is  necessary,  further,  for  an  exact  defi- 


MEASUREMENT  OF  CHEMICAL  ENERGY  5 

nition,  to  settle  on  some  one  temperature  from  which 
the  heating  of  the  water  i°  Cel.  shall  follow.  For  this 
purpose  we  can  choose  either  o°  Cel.  as  the  initial 
temperature,  in  which  case  the  unit  of  heat  is  known 
as  the  zero  calorie  (Nullkalorie);  or  we  can  choose  the 
one-hundredth  part  of  the  amount  of_heat  necessary 
for  the  heating  from  o°  to  100°  Cel.,  which  unit  is 
called  the  mean  calorie  (mittlere  Kalorie) ;  or,  finally, 
we  can  choose  the  most  convenient  temperature  for 
observation,  that  between  15°  and  20°  Cel.,  as  the 
initial  temperature  (Gebrauchskalorie).  For  the  defini- 
tion of  a  practical  unit,  the  one  proposed  by  Dieterici  * 
for  a  mean  calorie,  as  determined  with  an  ice-calorim- 
eter, is  to  be  recommended :  it  is  the  heat  necessary 
to  melt  such  an  amount  of  ice  at  o°  Cel.  as  to  cause  a 
decrease  of  volume  equal  to  that  of  15.44  mg.  of  mer- 
cury. The  small  differences  which  occur  in  the  sizes 
of  the  different  calories  are,  in  the  present  state  of 
thermochemical  measurements,  of  little  importance ; 
but  for  the  calculation  of  the  constants  they  must  be 
considered.  I  mean  calorie  =  1.0045  zero-calories. 

Since  the  differences  of  energy  in  thermochemistry, 
when  measured  in  any  of  the  above-defined  units,  are 
very  large,  and  are  not  exact  to  more  than  three  or 
four  hundred  units,  larger  ones  have  been  proposed. 
These  are  the  Ostwald  calorie,  K,f  and  the  large  calorie, 

*  Wied.  Ann.  33,  1888. 

f  Wied.  Ann.  33,  1888.  The  Ostwald  calorie,  K,  is  equal  to  100 
mean  calories  at  18°  Cel. ;  it  is  equal  to,  at  Baltimore,  4183  megergs. 
See  Ostwald,  Lehrbuch  d.  Allgemeinen  Chemie,  vol.  n,  i,  p.  74  (Leip- 
zig, 1893). — THE  TRANSLATOR. 


6     PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 

Cal.,  which  are,  respectively,  one  hundred  and  one 
thousand  times  as  great  as  the  above-defined  small 
calorie,  cal.  Between  these  units  we  then  have  the 
following  relations  : 

I  cal.  =  i  g.  °  Cel. 
I  K.  =  100  g.  °  Cel. 
i  Cal.  =  1000  g.  °  Cel. 

For  the  clear  and  complete  expression  of  the 
results  of  observations  on  chemical  energies  we  must 
still  find  proper  units  for  the  amounts  of  matter  in 
which  these  energies  are  stored.  Stoichiometry  has 
led — in  addition  to  the  knowledge  that  these  amounts 
are  proportional  to  the  masses  or  weights* — to  propor- 
tional numbers,  the  molecular  weights,  which  are  spe- 
cific properties  of  the  single  chemical  elements,  and 
are  expressed  by  the  chemical  formulae.  Chemical 
energetics  uses  the  same  symbols.  The  chemical  sym- 
bols of  substances  must  in  this  case  represent  as  many 
grams  as  are  expressed  by  the  molecular  weight ;  in 
other  words,  the  so-called  gram-molecule  or  mol  (Ost- 
wald).  It  is  easier  for  chemical  purposes  to  measure 
an  amount  of  water  in  this  way  than  in  grams;  the 
specific  constant  or  mol  being  here  18  grams,  so  that 
wH2O  is  equal  to  i$n  grams.  The  consequence  of 
this  is,  that  the  symbols  of  the  atoms  also  represent 

*  Weight  is  here  used  in  the  same  sense  as  mass,  therefore  differ 
ent  from  gravity.  The  original  conception  of  mass  established  its 
relation  to  kinetic  energy,  as  Ostwald  has  already  observed  ;  that  is, 
stands  also  in  relation  to  potential  and  to  chemical  energy  as  the 
result  of  experience. 


MEASUREMENT  0. 


certain  masses :  H  means  I  gram  of  hydrogen ;  O,  16 
grams  of  oxygen;  and  O2,  32  grams,  etc. 

The  chemical  symbols  in  energetics  represent,  in 
addition  to  the  specific  units  of  mass,  also  the  intrinsic 
energies  that  these  units  of  mass  contain.  That  18 
grams  of  fluid  water  contain,  at  the  same  temperature, 
less  energy  than  2  grams  of  hydrogen  plus  16  grams  of 
oxygen,  from  which  it  is  formed,  and  so  much  less  as 
is  represented  by  68000  cals.,  can  now  be  shown  by 
the  formula 

2H  +  O  =  H3O  +  68000  cals. 

* y '         liquid 

gaseous 

or 

2H  +  O  —  H2O  =  68000  cals. 

L 

The  appearance  of   an   amount  of   energy  in  the 

formula  shows  us  that  here  the  chemical  symbols  are 
used  in  the  second  meaning  also,  and  not  only  as  a 
number  of  units  of  mass. 

If  by  the  combustion  of  hydrogen  (in  a  calorimetri- 
cal  bomb)  only  heat  is  developed,  and  this  heat  is  not 
used  up  in  altering  the  temperature  of  the  products  of 
the  reaction,  then  by  the  formation  of  every  18  grams 
of  water  68  cals.  of  heat  will  be  liberated.  If,  on  the 
other  hand,  some  of  the  energy  is  developed  in  another 
form,  then  the  amount  of  heat  is  the  same  as  it  was 
before,  less  the  amount  of  heat  which  the  other  form  of 
energy  corresponds  to.  The  formula  as  given  means 
only  that  the  difference  of  energy  between  the  states 


8      PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

2H  +  O  and  HaO  is,  for  every  18  grams  of  the  latter 
formed,  680  K. 

The  explanation  of  all  the  conditions  to  be  observed 
in  the  formation  and  use  of  equations  of  chemical 
energy  is  the  object  of  the  next  chapter,  and  of  some 
of  those  following. 

It  will  be  well,  however,  to  state  here  the  effect 
which  the  differences  in  the  atomic  weights  have  upon 
the  values  of  .the  differences  of  energy.  If  we  assume 
O  =  16,  then  the  most  probable  value  for  H  will  be 
1.0032;  by  a  small  calculation  we  find  that  the  differ- 
ence  in  energy,  between  water  and  its  components,  by 
these  values,  is  24  cals.  larger;  and  if  we  choose  H  =  I, 
O  =  15.95,  then  the  difference  will  be  189  cals.  smaller  ; 
than  that  value  given  in  the  equation. 

It  may  be  well  to  remark  here  that  the  amount  of 
heat  which  expresses  the  difference  of  energy  for  a 
gram-molecule  or  mol.  is  called  the  heat  of  reaction  by 
constant  volume.*  It  is  reckoned  positive  when  the 
intrinsic  energy  of  the  reaction  decreases,  and  is  called, 
according  to  the  chemical  or  physical  nature  of  the 
reaction,  the  heat  of  aggregation,  solution,  modification, 
dilution,  or  combustion.  The  study  of  the  determina- 
tion of  these  differences  of  chemical  energy  is  known 
as  Thermochemistry. 

It  will  be  well,  perhaps,  to  append  here  a  few  cal- 
culations, which  are  possible  from  the  few  principles  of 

*  Here  in  the  calorimetrical  bomb  the  volume  is  constant,  and 
hence  the  above  designation.  In  case  of  H  burning  in  open  air, 
under  atmospheric  pressure,  the  amount  of  energy  would  be  different 
from  this,  as  will  be  explained  later. — TRANS. 


MEASUREMENT  OF  CHEMICAL  ENERGY  9 

thermochemistry  that  have  already  been  explained, 
and  thus  to  make  the  student  familiar  with  their  appli- 
cation. 

If  H2SO4  unites  with  «H2O,  we  have,  according  to 
J.  Thomsen,  an  amount  of  heat  equal  to : 


I  17860  cals. 


This  formula,  from  experiment,  shows  small  varia- 
tion, except  for  medium  concentrations,  between,  for 
example,  n  =  50  and  n  =  1000.  How  much  heat,  R, 
will  be  developed,  according  to  this,  by  the  admixture 
of  ;t:H2SO4  \vith  ;/zH2O,  providing  that  the  change  of 
volume,  by  the  mixing,  brings  with  it  no  noticeable 
change  in  the  difference  of  energy? 

R  is  plainly  x  times  as  great  an  amount  of  heat  as 

iH2SO4  with  —  HaO  would   liberate;    therefore,  since 

% 

m 

-  =  n, 


m 


.  T  7860  =  —  -2?—       .  17860  cals. 
m  4-  1  .798^-       ' 


What  amount  of  heat  will  be  produced  by  the  addi- 
tion of  the  very  small  quantity  dx  of  the  acid  to  the 
mixture  of  x  .  H2SO4  and  m.  H2O  ;  and,  on  the  other 
hand,  what  amount  by  the  addition  of  the  very  small 
quantity,  dm,  of  water  ? 


10  PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 
By  differentiating,*  we  find 

**dx=7  -. 17*60.  dx 

•dx  (m  +1.789*) 

.  17860.^. 


(n  +  1.798) 


'dm  (m  +1.798*)' 

=  /         f^Q^  178ft).  <**• 

(n  +  1.798) 

Up  to  what  dilution  does  a  drop  of  H2SOi  bring  a 
greater  amount  of  heat  than  a  drop  of  H2O  of  the 
same  weight  ? 

If  both  drops  weigh  the  same,  let  us  say  dM  grams, 
then  the  drop  of  H2SO4  contains 

dM 

=  ~$f 

gram-molecules-  the  drop  of  water,  however,  contains 

dM 


gram  molecules.      Now  ^—  dx  <  —  dm,  according   as 

fi*^  1.798  dm,  or 

a>  1.798  X98 
a 


*  Horstmann,    Theor.  Chemie   in   Graham-Otto's    Lehrbuch,   II, 
1885. 


MEASUREMENT  OF  CHEMICAL   ENERGY.         II 

If  one  adds  to  a  mixture  of  x  .  HaSO4  and  m  .  H2O, 
dx  .  H2SO4  and  dm  .  H2O  at  the  same  time,  then 

dm  :  dx  —  m  :  x  =  n, 

or  no  other  development  of  heat  than  that  caused  by 
the  mixing  of  the  small  portions  dm  and  dx. 

How  much  heat,  Q,  is  formed  when  the  dilute  acid 
HC1.//H2O  is  added  to  an  unlimited  amount  of  water? 
From  J.  Thomsen  we  have 

HC1H20  +  0-  i)H20 

=  HCLHaO,(»- 


and  it  follows,  when  we  substitute  for  n  its  value  co  , 
that 

HC1.H20  +  Aq  =  HC1.HA  Aq  +  1  19.8  K. 

On  the  other  hand,  the  amount  of  heat,  Q,  must 
satisfy  the  equation 

sO  +  (»-  i)H,O 

=  HCl.HAAq+       -rn9.8+<2, 


where  we  can  calculate  (n  —  i)HaO  as  Aq. 

By  subtracting  the  one  from  the  other  we  find 


7          n 

while  Berthelot*  found  by  experiment 


*  See  Planck,  Grundriss  der  allg.  Thermo-chemie  in  Ladenburg's 
Hand-worterbuch  der  Chemie,  1893.  Also  published  separately  as  a 
sma.l  book.—  TRANS. 


12   PRINCIPLES   OF  MATHEMATICAL   CHEMISTRY. 

The  foregoing  examples  sho^v,  by  their  application 
of  the  principle  of  the  conservation  of  energy,  without 
further  aids,  that  all  reactions  or  series  of  reactions  that 
transform  one  substance  into  another  must  liberate  the 
same  amount  of  energy,  which  is  the  difference  of  the 
intrinsic  energies  of  these  substances.  If  they  lead,  on 
the  one  hand,  to  the  confirmation  of  the  principle  of 
the  conservation  of  energy,  they  lead  also,  on  the  other, 
to  the  determination  of  the  energy  differences  of  reac- 
tions which  are  not  directly  accessible  to  observation; 
especially,  in  many  cases,  to  the  difference  of  energy 
between  a  substance  and  its  elementary  constituents, 
the  so-called  heat  of  formation  (Bildungswarme).  Since 
from  the  difference  of  energy  between  a  compound  and 
its  elements  we  can  find,  by  subtraction,  the  energy 
of  the  compound  (i.e.,  the  difference  between  its  state 
and  the  standard  state),  thermochemistry  reaches  its 
greatest  value,  as  far  as  concerns  chemical  equations 
proper,  by  the  determination  of  the  heats  of  formation. 

Thus,  for  example,  from  the  heats  of  formation 
and  of  solution  of  magnesium  chloride  and  of  sodium 
chloride, 


Mg  +  Cl,  +  Aq  =  MgCl2Aq  +  187  CaL, 

Na  +  Cl  =  NaCl  +  98  CaL, 
Na  +  Cl  +  Aq  =  NaClAq  +  97  CaL, 

we  can  find  the  heats  of  reaction,   Q  and  Q',  of  the 
chemical  process  between  MgCl2  and  Na. 
If  we  substitute  in  the  equation 

MgCla  +  2Na  =  2NaCl  +  Mg  +  Q, 


MEASUREMENT   OF  CHEMICAL   ENERGY.         13 

the  values  for  MgCl2  and  NaCl,  as  found  in  the  first 
and  third  equations,  we  obtain 


Mg-f  C12-  151 

or         Q  =  45  Cal. 

In  the  same  way  we  find,  for  the  case  that  the 
reaction  takes  place  in  a  water  solution, 

Q  =  7  Cal. 
From  thermochemical  data, 

C  +  20  =  C02  +  97  Cal.; 
CO  +  O  =  C02  +  68  Cal.; 

therefore,  by  subtraction,  we  find  the  heat  of  forma- 
tion of  carbon  monoxide, 

C  +  O  =  CO  +  29  Cal. 

In  this  case  amorphous  carbon  is  to  be  understood 
by  C.  If  one  wished  to  find  an  equation  for  one  of 
the  other  modifications  of  carbon,  the  amount  of  en- 
ergy would  have  to  be  changed,  for  the  intrinsic  ener- 
gies of  the  elementary  elements  would  change.  One 
mol  of  amorphous  carbon,  of  graphite,  and  of  diamond 
give,  respectively,  by  combustion, 

97650,     94810,     94310  cals., 

which  shows  a  difference  of  intrinsic  energy  of  more 
than  3000  cal.  for  amorphous  carbon  and  diamond. 
J.  Thomsen  designates  the  heat  of  formation  by  a 


14  PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 

symbol,  so  that,  for  example,  the  contents  of  the  above 
equation  for  MgCl2  would  take  the  form 

(Mg,Clf)=  151  CaL, 

where  the  parenthesis  means  that  the  chemical  ele- 
ments which  are  separated  by  the  comma  unite  to 
form  a  compound.  The  symbol  of  Thomsen's  also 
means,  when  used  for  constant  volume,  the  excess  of 
energy  of  the  elements  over  the  compound. 


CHAPTER  III. 

CHANGES  IN  TEMPERATURE  AND  IN  THE  STATE 
OF  AGGREGATION. 

THE  energy  of  a  substance  changes  by  each  change 
of  a  chemical  or  physical  nature  which  takes  place  in 
it.  Therefore  thermochemical  equations,  strictly  con- 
sidered, should  be  given  not  only  in  a  special  chemical 
state,  but  also  in  a  special  physical  one,  under  which 
each  of  the  reactions  of  the  elementary  substances 
takes  place.  We  must  know  the  pressure,  tempera- 
ture, aggregation,  and  allotropic  state  of  eaclT  of  the 
bodies  which  come  under  our  consideration.  As  self- 
evident,  when  not  otherwise  expressly  stated,  it  is  to 
be  understood  in  all  thermochemical  equations  that 
the  substances  are  in  the  standard  state  (Normalzu- 
stand),  i.e.,  at  o°  Cels.  temperature  and  under  the 
pressure  of  I  atmosphere,  and  with  no  differences  of 
intensity  (e.g.,  electrical)  existing. 

The  state  of  aggregation  of  a  body  is  best  distin- 
guished by  certain  signs,  as  by  different  letters  (Ost- 
wald)  or  by  brackets  and  parentheses,  so  that  H2O 
means  steam,  (HaO)  liquid  water,  and  [H2O]  ice.  The 
specific  heats  necessary  for  providing  for  changes  of 
temperature  are,  as  are  also  the  latent  heats  and  heats 


1  6  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

of  aggregation,  based  generally  in  physics  on  the  unit 
of  mass,  the  gram  ;  but  in  thermochemistry  they  must 
be  recalculated  and  based  upon  the  atomic  and  molec- 
ular weights.  For  instance,  if  c  is  the  specific  heat  of 
a  substance  and  m  its  molecular  weight,  then  me  is 
the  molecular  heat,  and  ac  the  atomic  heat  if  a  is  the 
atomic  weight. 

These  general  rules  will  be  illustrated  sufficiently 
by  the  following  examples  : 

From  Julius  Thomsen's  observations  for  H  —  i  and 
O  =  16,  or  H2O  =  1  8,  we  have  the  equation 

2H  +  O  =  (HaO)  +  67484  cals.  ;       18°  Cels.,  i  atmos- 
phere pressure. 

What  would  the  equation  be,  if,  instead  of  the  tem- 
perature of  observation,  18°  Cels.,  we  had  any  other, 
$°,  still  retaining  the  volume  as  it  was  at  1  8°  ? 

The  specific  heats*  of  hydrogen,  oxygen,  and 
liquid  water  are  expressed,  for  constant  volume,  by 
the  equations 

-  1  8); 


18° 

=  0  +  2.48(S-  18); 

18° 

=  (H20)+i8(fl-i8). 

18° 


*  These  and  the  following  corresponding  data  are  taken  from 
Regnault's  observations,  as  collected  by  Zeuner  in  his  Technischer 
:Thermodynamik. 


CHANGES  IN   TEMPERATURE,  ETC.  I/ 

From  these  we  obtain 

+-H.O)  -  HH-0  -  (H.O)  -  107(0  -  18) 


=  67484  —  10.7(0  —  1  8) 
=  67677  —  10.  71^, 
or 

r  Temp.  =  £°.    Vol. 

Ha+  O  =  (H,O)  +  67677  -  io.;0  \       same  as  for  18° 

(       Gels. 

How  great  is  the  difference  of  energy  between  dry 
steam  at  $°  and  its  component  gases  H2  and  O  at  o°  ? 

We  start  from  the  specific  heats  at  constant  volume, 
and  the  energy  differences,  as  given  by  Zeunert  be- 
tween fluid  water  at  $°  and  steam  at  the  same  tem- 
perature. We  have 

H8=  H2-4.82X  18; 

o°          1  8° 

O  =  O  -  2.48  X  18; 

o°        18° 


V  18° 

HaO  =  (H,0)  +  18(575.40  -  0.791*). 

&°  O° 

By  subtracting  the  last  two  equations  from  the  sum 
of  the  first  two,  after  eliminating  the  energy  (H2O),  we 
find 

H2  +  O  -  H20  =  H2  +  O  -  H20  -  18(564.7  +0.209^) 

o°  18° 

=  67484  —  1  8  X  564.7  —  1  8  X  0.209$ 


or 


1  8   PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 

By  this  equation  the  question  is  answered,  but  only 
so  when  the  specific  heats  are  considered  as  independ- 
ent of  the  temperature. 

It  is  possible  to  prove  that  in  general  the  heat  of 
formation  is  a  function  of  the  temperature  of  the  same. 
(Kirchhoff.)  If  the  intrinsic  energies  of  the  unit  of 
mass  of  two  substances  at  the  temperature  $0°  are  Et 
and  £a,  and  these  substances  unite  in  the  amounts  Ml 
and  M^  at  the  temperature  £°,  at  the  same  time  setting 
free  the  amount  of  heat  Q-  and  if  after  the  reaction, 
when  the  temperature  is  again  $0°,  the  intrinsic  energy 
for  the  unit  of  mass  is  E,  —  then  by  the  principle  of  the 
conservation  of  energy  we  have 

(M, 

-Q+ 
«A 


where  clt  c^  and  c  are  the  specific  heats,  for  constant 
volume,  of  the  substances  and  the  product  respectively. 
If  we  allow  the  temperature  $°  to  vary,  then 


which  is  in  general  not  equal  to  zero,  i.e.,  Q  will  in 
general  vary  also.  For  the  combustion  of  H,  when  Q 
refers  to  one  mol,  we  have,  as  already  given  above, 

a<2 

=  ~  I0'7' 


CHAPTER  IV. 

MECHANICAL   ENERGY. 

ALTHOUGH  heat  is  the  form  of  energy  which  is  pro- 
duced in  the  greatest  amount  by  the  chemical  energy 
differences,  still  it  is  impossible,  even  in  the  most  impor- 
tant cases,  to  shut  out  entirely  the  other  forms.  It  is 
necessary  first  for  us  to  study  the  changes  of  volume 
in  chemical  processes  to  find  what  amounts  of  energy 
they  represent,  and 'to  do  this  we  must  consider  all  the 
phenomena  of  motion. 

As  far  as  concerns  the  motion  of  solid  bodies,  we 
must  consider  two  forms.  The  kinetic  energy,  or  the 
energy  of  motion  of  a  body  whose  mass  is  mg  and 

.     .      .      cm.   . 

whose  velocity  is  c ,  is 

sec. 

1  8  g.cm.a 

-  me*  -  — 5-  • 

2  sec. 

Its  unit  is  I  g.cm.'sec."2,  and  is  called  the  Erg.  If 
the  body  is  located  in  a  system  of  co-ordinates,  and 
they  change  in  the  time  dt  by  dx,  dy,  and  dz,  we  can 
express  the  kinetic  energy  by  the  following  equation: 


2O  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

Each  change  in  the  kinetic  energy  of  a  body  is 
called  work  ;  for  example,  the  mechanical  work  along 
the  x  co-ordinate  in  the  time  dt  is 

dx  d*x  ,  d"x  , 

m  —  —  -  dt  =  m  —  dx. 
dt  df  df 

According  to  the  principle  of  the  conservation  of 
energy,  each  change  of  this  sort  is  an  exchange  of 
energy;  and  this  has  led  to  the  conception  of  another 
form  of  energy,  which  acts  as  a  source  and  an  outlet 
to  the  kinetic  energy.  This  form  is  called  the  potential 
energy. 

The  characteristic  property  of  each  potential  en- 
ergy is  this,  that  its  amount  depends  only  upon  the 
position  of  the  movable  body.  On  this  account  pote-n- 
^ial  energy  has  been  well  called  the  energy  of  position. 
If  the  position  of  a  body  is  changed,  with  respect  to  its 
co-ordinates,  by  the  amount,  dx,  dy,  and  dz,  then  the 
total  change  of  its  kinetic  energy  is  given  by  the  three 
parts  Xdx,  Ydy,  and  Zdz,  and  the  factors,  X,  Y,  and  Z 
are  the  components  of  a  vector  which  is  the  force 
which  acts  upon  the  mass.  The  unit  of  force  is  the 
Dyne,  or  i  g.cm.sec."5 

The  comparison  of  the  expressions  of  mechanical 
work  show  that 


or  Force  =  Mass  X  Acceleration. 

Of   the    two  factors  which   are   necessary  for  the 
measurement  of  kinetic  and  potential  energy,  or  for 


MECHANICAL  ENERGY.  21 

the  measurement  of  kinetic  energy  and  its  changes, 
besides  those  of  space  and  time,  the  one  is  determinable 
from  the  other  when  the  acceleration  is  known.  It  is 
especially  easy  from  the  above  to  find  the  amount  of  a 
force  when  we  know  the  mass  and  the  acceleration.  In 
the  same  way  we  could  determine  the  mass  if  we  chose 
a  unit  of  force  to  start  with.  In  technology  this  latter 
method  is  preferred.  Here  neither  the  mass  nor,  what 
is  practically  the  same,  the  weight  of  I  cubic  centi- 
meter of  water  is  chosen  as  the  unit  of  force,  but  the 
acceleration  of  this  body,  due  to  gravitation,  as  is 
shown  in  some  certain  place  on  the  earth,  the  standard 
place.  The  units  derived  from  gravitation  are  distin- 
guished by  stars,  *,  from  the  units  of  mass.  Thus  we 
have 

(  Acceleration  of  \ 

Unit  of  force  =  Unit  of  mass  X  •<    gravitation  in  the  >  , 

(    standard  place       ) 

or  Grams  X  Gravitation  —  Grams  X  Mass  X  981  — -0. 

sec.3 

g*  =  981  g.cm.sec.-8 
==  981  dynes. 

The  corresponding  amount  of  energy  is 
Grams  X  Gravitation  X  cm.  =  g.*cm.  =  981  ergs, 

and  is  the  loooooth  part  of  the  kilogrammeter  (properly 
kilogram  X  gravitation  X  meter)  Kg*m,  which  repre- 
sents the  energy  that  must  be  expended  to  raise  I  kilo- 
gram i  meter  in  the  air. 

The  energies  of  position  which  occur  in  Nature  are? 
divided  (Ostwald)  into  distance,  surface,  and  volume! 


22   PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 

energy.  In  the  first  case  the  changes  of  energy  are 
proportional  to  the  changes  of  distance  between  a  pair 
of  bodies,  and  they  exert  either  an  attracting  or  a  re- 
pelling force.  Gravitation  and  several  kinds  of  electric 
and  magnetic  energies  belong  to  this  group.  The  sec- 
ond group,  as  was  the  first,  is  also  of  little  importance 
in  our  present  work,  for  surface  energy  is  proportional 
to  the  surface,  which  is  in  a  state  of  tension. 

On  the  other  hand,  the  volume  energy  is  of  ex- 
treme importance  in  thermochemistry,  because  it  can 
be  transformed  directly  into  kinetic  energy,  and  also 
into  heat.  Liquid  and  gaseous  bodies,  which,  on 
account  of  their  simplicity,  only  will  be  treated  here, 
show  the  property  that  when  an  increase  of  volume, 
and  no  other  change,  takes  place  the  intrinsic  energy 
decreases.  The  proportion  of  this  decrease  of  energy 
—  dEv  to  the  increase  of  volume,  dV,  is  placed  equal 
to  the  pressure/,  so  that 

dEv 
~dV=P> 

and  therefore 

T,   .      .  Unit  of  energy          g  g* 

Unit  of  pressure^  — — — - — --=£  =  — * — .     or    £—, -t. 
Unit  of  volume     cm.  sec.  cm. 

The  pressure,  so  defined,  can  also  be  considered — 
and  this  is  the  usual  way — as  a  force  which  does  work 
against  resistance.  We  imagine  the  surface  of  the 
liquid  or  gaseous  body  under  consideration  as  immov- 
able except  in  the  small  area  of  the  size  q  sq.  cm.  A 
small  increase  of  volume  is  then  only  possible  by  a 
movement  outward  of  this  area  in  the  direction  of  its 


MECHANICAL  ENERGY.  2$ 

normal,  let  us  say  of  the  amount  dn  (Fig.  i),  so  meas- 
ured that  dV  —  q.dn.     If  there  is  a  force,  P,  directed 
downward  in  the  direction   of   the  normal  of 
the  surface  which  works  against  this  outward 
pressure,   then  during  the    expansion  dV  the 

mechanical  work  done  will  be  Pdn.     From  the 

•     •  i       r  ^i  r  FlG- 

principle  of  the  conservation  of  energy 

dE,=  -pdV  =  - 
but  dV  '=  —  qdn,  and  therefore 


i.e.,  the  pressure  can  be  considered  as  the  force  neces- 
isary  to  stop  entirely  the  increase  of  volume. 

It  is  now  only  necessary  to  give  the  relation  be- 
tween the  calorie,  as  described  in  a  previous  chapter, 
and  the  erg  or  the  technical  kilogrammeten  From 
Joule's  very  careful  experiments, 

I  Cal.  =  423.55  tech.  kilogrammeters  ; 

I  cal.  =  42355  X  981  ergs  =  415.5  X  io5  ergs. 

The  latest  experiments  by  Dieterici,  by  which  the 
mechanical  equivalent  of  heat  was  determined  from 
the  heat  developed  by  the  electrical  current,  give  the 
value  of  the  practical  mean  calorie  for  mean  lati- 
tudes as 

i  cal.  =  424.  36  X  IOB  ergs, 

i  Cal.  =  432.5  tech.  kilogrammeters, 

f  Counted  negative  because  done  by  the  fluid,  and  consequently 
lost  by  it;  if  from  the  outside,  thus  compressing  the  fluid,  it  would  be 
positive,  for  then  it  would  be  added  to  that  of  the  fluid.  —  TRANS. 


24  PRINCIPLES  OF   MATHEMATICAL    CHEMISTRY. 

with  a  probable  error  of  0.17  for  the  number  424.36. 
The  exact  recalculation  of  the  earlier  results  of  Joule, 
by  help  of  the  data  of  Dieterici,  is  difficult,  for  the 
reason  that  the  conception  of  i°  Gels,  is  uncertain; 
that  is,  the  reduction  of  the  readings  on  the  mercury- 
thermometer  to  the  air-thermometer. 


CHAPTER  V. 

THE   VOLUME   ENERGY   OF  GASES. 

FOR  use  in  thermochemistry,  we  need  the  mechani- 
cal considerations  treated  of  in  the  last  chapter  only 
so  far  as  to  be  able  to  bring  into  calculation  the  vol- 
ume energy  of  gases  which  are  present  in  chemical 
reactions. 

As  long  as  a  body  remains  solid  or  liquid  it  under- 
goes such  a  small  change  of  volume  that  the  changes 
of  energy  can  be  neglected  when,  as  usual,  the  reaction 
takes  place  under  ordinary  pressures.  Even  in  the 
case  of  gases,  where  these  changes  of  volume  are  so 
large,  the  corrections  are  so  unimportant,  in  the  pres- 
ent state  of  thermochemical  methods,  as  to  affect  only 
the  last  certain  figure  of  the  results. 

Gaseous  bodies,  when  brought  sufficiently  high 
above  their  critical  states,  follow  the  equation  of  state, 

(i)  pv  =  RO, 

in  which  /  is  the  pressure,  v  the  specific  volume,  i.e., 
the  volume  of  I  gram  of  gas,  6  the  absolute  tempera- 
ture (273°  higher  than  the  Celsius  temperature  $),  and 

25 


26  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

R  a  constant  peculiar  to  the  gas.     The  absolute  tem- 
perature is 

(2)  e  =  273  +  a. 

This  conception  of  absolute  temperature  is  intro- 
duced in  order  to  bring  into  use  in  calculations,  in  the 
most  convenient  way,  the  fact  that,  under  constant 
pressure,  when  the  temperature  is  raised  i°  Cels.,  all 
gases  expand  1/273  of  the  volume  which  they  occupy 
at  0°  Cels. 

If  a  gas  whose  mass  is  M  occupies  the  volume  V 
its  specific  volume  is 

V  cm.3 


(3) 


M    g 


and   its  specific  weight  (the  reciprocal  of  its  specific 
volume)  is 


v       V  cm.8' 
From  these  we  find 

(4)  pV^R.M.B. 

When  we  have  one  mol  of  gas,  and  m  is  its  molec- 
ular weight,  then 

pvm  =  R  .  m  .  0, 

(5)  or      pV.  =  R.O. 

Here  VQ  =  vm,  the  volume  of  I  mol  (or  the  molec- 
ular volume  in  cubic  centimeters),  and  since,  by  Ava- 
gadro's  and  Gay-Lussac's  law,  this  is  the  same  for  all 


THE    VOLUME   ENERGY  OF  GASES.  27 

gases,  under  the  same  pressure  and  temperature,  it 
follows  that 

(6)  R0  =  R.m, 

which  has  one  and  the  same  value  for  all  gases. 

R  could  be  called  the  specific,  and  R0  the  general, 
gas  constant. 

These  important  constants  can  therefore  be  deter- 
mined for  any  gas  of  known  molecular  weight,  in  any 
state,  from  its  pressure,  volume,  and  temperature.  We 
choose  32  grams  of  oxygen  in  the  standard  state, 
and  place  its  atomic  weight  at  16.  Since  the  specific 
weight  of  oxygen  is  0.00143  gr.  cm.~3,  its  molecular 
volume  is 

—  -  -  =  22400  c.cm., 
0.00143 

and  this  is  the  volume  of  one  mol  of  any  perfect  gas. 
With/—  I  atmosphere  =  76x  13.596=  1033.3  g.*cm.2 
and  fl  =  o°  Cels.  or  6  =  273°  Cels.,  it  follows  that 


=  84800  =  0.848 


=  832  X  io6  ergs 

_   848    cals.  _         ,  cals. 
~  ^S  ?C.  ~  ^C/ 

or  R0  =  2  cals.,  nearly. 

To   prove   the   proposition   that   R9   has   the  same 
value  for  all  gases  in  all  states,  we  will  calculate  it  for 


28   PRINCIPLES  OF   MATHEMATICAL    CHEMISTRY. 

saturated  water  vapor  at  o°.     Dieterici  f  gives  for  this 
case 

/  =  4.619  mm.  Hg  ; 

cdm. 


v  =  204.680 
s  —  4.8856 


g 


dm.3 

We  have  then,  since  the  molecular  weight  is  18, 
_  0.4619  X  13-596  X  204680  X  18  _  Q^^g*cm. 

273  75    i°c' 

which  corresponds  closely  with  the  calculated  result 
given  above. 

f  With  these  few  words  of  preparation  we  can  now 
enter  into  the  calculation  of  the  changes  of  volume 
energy  which  take  place  in  chemical  reactions.  If  a 
substance  gives  off  a  small  amount  of  gas,  <^Fccm.,  while 
the  pressure  (e.g.,  the  atmospheric  pressure)  amounts 
to  /  dynes  per  square  centimeter,  then  its  intrinsic 
energy  decreases  by  the  amount  necessary  to  overcome 
this  pressure  ;  that  is,  the  intrinsic  energy  increased  by 
the  amount  —  pdV.  The  mechanical  work  that  is  de- 
veloped by  a  chemical  reaction,  by  which,  under  con- 
stant pressure,  M  grams  of  gas  is  formed,  is  therefore, 
by  the  equation  of  state  of  gases, 


or  the  generation  of  I  mol  or  m  grams  of  gas  is  equal 
to  RJ. 

f  Wied.  Ann.  38,  1889. 


THE    VOLUME   ENERGY  OF  GASES.  2Q 

Just  so  often  as,  by  the  absolute  temperature  6,  one 
mo  I  of  gas  is  generated  or  dissipated  at  any  unvarying 
constant  pressure,  just  so  often  is  the  intrinsic  energy  of 
the  substance  decreased  or  increased  by  Rfl,  i.e.,  by 
nearly  26  calories  (cals.}. 

This  law  makes  it  possible  for  us  to  determine  the 
energy  differences,  without  keeping  the  volume  con- 
stant, by  allowing  the  reaction  to  proceed  under  atmos- 
pheric pressure.  If  a  reaction  takes  place,  for  ex-ample, 
a  combustion  under  atmospheric  pressure,  and  if  the 
place  of  combustion  is  surrounded  by  a  calorimeter, 
so  that  the  gas  escaping  from  it  takes  with  it  no  more 
heat  than  is  necessary  to  bring  it  to  the  same  tempera- 
ture as  the  outer  air,  then  there  will  be  work  developed 
or  used  up  by  the  air-pressure,  and  the  differences  of 
intrinsic  energy  will  not  come  to  observation  as  heat 
alone,  but  also  as  work,  from  which,  however,4n  addi- 
tion to  the  heat  by  the  above  law,  we  can  calculate  the 
energy  differences.  The  heats  of  reactions  by  constant 
pressure  are  more  often  observed,  and  are  of  greater 
importance  for  practical  use  of  the  theory  than  those 
by  constant  volume — so  much  so  indeed  that  it  is  cus- 
tomary to  place  them,  instead  of  the  others,  in  thermo- 
chemical  equations.  We  find,  for  example,  general 
data  such  as  this:  ^ 

(  H,  +  0  =  (H,0)  +  68357  cals. 

(  at  1 8°  Cels.,  constant  pressure  of  I  atmosphere 

In  this  equation,  however,  the  chemical  symbols  are 
used  in  still  a  third  sense,  besides  the  two  already  men- 
tioned on  pages  6  and  7, 


30  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

There  it  was  explained  that  the  symbols  mean 
molecular  or  atomic  weights,  when  the  equations  in 
which  they  occur  contain  no  values  of  energy;  on  the 
other  hand,  when  these  are  present  the  symbols  mean 
the  amounts  of  intrinsic  energy  which  the  gram  molec- 
ular or  gram  atomic  weights  contain.  That,  in  the 
above  equation,  the  symbols  have  a  third  meaning  is 
apparent,  for  otherwise  it  would  stand  in  contradiction 
to  the  one  on  page  16.  What  this  third  meaning,  is  can 
be  shown  in  the  following  manner: 

If  E  is  the  total  intrinsic  energy  of  the  reacting 
substances  in  any  moment  of  the  chemical  process,  and 
/  the  pressure  at  the  time  ;  and  if  in  an  element  of 
time  the  energy  increases  by  the  amount  dE  and  the 
total  volume  by  the  amount  dV,  and  the  heat  dQ  is 
absorbed,  —  then,  according  to  the  principle  of  the  con- 
servation of  energy, 

dE  =  dQ-  pdV.     -    K 

If  the  volume  does  not  change,  then  the  heat  ab- 
sorbed is  the  differential  of  the  heat  of  reaction  Qv,  by 
constant  volume,  or 


i.e.,  the  difference  of  the  intrinsic  energies  is  measured 
by  the  heat  developed,  when  the  volume  is  constant. 
If,  however,  the  volume  changes  (so  that  dV  is  no  longer 
equal  to  zero),  but  the  pressure  /  remains  constant, 
then  pdV—d(Vp)',  therefore  the  differential  of  the 
heat  of  reaction  QP  by  constant  pressure  satisfies  the 
equation 

dQ; 


THE    VOLUME 

i.e.,  the  development  of  heat  by  constant  pressure 
measures  the  decrease  that  the  quantity  E  +  pV  suf- 
fers during  the  reaction  :  this  we  will  call  the  free  energy 
by  constant  pressure.  It  is  the  intrinsic  energy  increased 
by  the  product  of  pressure  and  volume. 

The  addition  to  the  above  equation  of  the  words 
by  "  constant  pressure  "  means,  therefore,  that  by  the 
chemical  symbols  we  are  to  understand  the  free  energy 
by  constant  pressure  which  the  gram  molecular  or 
gram  atomic  weights  contain.  Since  for  one  mol  the 
product  pV •=.  pvm  =  R6m  —  R06,  or  nearly  20  cals., 
the  above  equation  is  transformed,  when  the  chemical 
symbols  are  to  mean  only  the  intrinsic  energies  con- 
tained in  the  mols,  into  the  following  form : 


H,  +  i.  R0^  +  0  +  t.R^=(H10)  + 68357  cals.; 
H2  +  O  =  (H20)  +  68357  -  i*  -  R0#  cals. ; 
H2  +  O  =  (H20)  +  68357  -  ii  .  2  .  ^cals.; 
=  (H2O)  +  67484  cal.  at  18°  Cels  ; 

in  which  form  it  corresponds  to  the  one  given  on  page 
16.  One  can  transform,  therefore,  equations  given  "  by 
constant  pressure  "  into  equations  referring  to  intrinsic 
energies,  by  adding  to  the  right  side  26  times  the 
increase  of  gaseous  mols  which  are  formed  during  the 
reaction. 

Upon  the  ground  of  the  preceding  explanation  we 
will  solve  some  of  the  problems  for  the  case  of  constant 
pressure  which  were  treated  under  constant  volume. 

Corresponding  to  the  question  treated  on  page  16, 
we  will  first  calculate  the  heat  that  is  developed  when 
oxyhydrogen  gas  at  $°  is  transformed  into  liquid  water 


32   PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

of  the  same  temperature  under  constant  pressure.    We 
start  from  Thomsen's  observation  that 
H2+O  =  (H2O)+68357  cals.  at  18°  Gels.,  I  atmosphere, 
and  use  Regnault's  results,  under  constant  pressure  ; 

H2  =  H2  +  6.82(0  —  1 8)  const,  pressure  ; 

O°          18° 

O=O  +3.48(0—  1 8)      " 

$°          1 8° 

It  follows  then,  as  before,  that 
H2+O=:  (H2O)  +  6S496  -  7.7$;  0°  and  I  atmosphere. 

Further,  we  can  find,  as  we  did  on  page  16,  the  heat 
which  is  developed  when  overheated  steam  at  0°  is 
formed  from  oxyhydrogen  gas  at  o°  and  atmospheric 
pressure.  We  need  for  that  the  so-called  total  heat,  A, 
of  steam,  i.e.,  the  heat  that  is  necessary  to  change  water 
at  o°  into  superheated  steam  at  0°,  under  constant 
pressure.  According  to  Zeuner  (and  others)  we  have 
from  Regnault's  observations,* 

H2O  =  (H,O)  +  1 8  (606.5  +  0.3050'  +  0.4805(0  -  O1), 

superheated       o° 
$° 

where  O1  is  the  temperature  of  dry  saturated  vapor,  at 
IOO°,  under  same  pressure.  ^We  obtain  then 

H2O  —  (H8O)  +  10602  +  8.6490,  const,  pressure. 

superheated         o° 
O° 

If  we  subtract  this  equation  from 

tr     i   c\      /w  c\\   i   *Q,^  j  °°  Cels.,  I  atmosphere  ) 
H2+()  =  (H30)  +  68496|      constant  pressure      ]' 

*  According   to    Dieterici's    measurements   a   few  corrections  are 
necessary.     In  place  of  606.5  he  finds  596,86, 


THE    VOLUME  ENERGY  OF  GASES.  33 

we  find 

(  under  constant  ) 

H3  +  O  =  H2O  +  57894  -  8.649  £  \    pressure   of  I  L 
o  superheated  (    atmosphere      ) 

By  this  we  obtain  the  answer  to  the  following  ques- 
tion :  To  what  temperature  could  the  oxyhydrogen 
flame,  under  atmospheric  pressure,  be  brought,  if  by 
the  combustion  only  overheated  steam  was  formed  and 
this  did  not  again  dissociate  ;  and  to  what  temperatures 
can  the  Regnault's  constants  be  used  ? 
The  answer  is 


8;694 

Since  100  grams  of  atmospheric  air  contains  23 
grams  of  oxygen  and  77  of  nitrogen,  or  to  16  grams  of 
oxygen  in  the  air  there  are  53.6  grams  of  nitrogen,  it 
is  necessary  that  the  57894  calories  (cals.)  developed  be 
used  also  to  heat  the  nitrogen.  The  specific  heat  of  N 
is  0.2438,  so  that  under  this  condition  the  temperature 
of  combustion,  fy,  of  hydrogen  in  the  air  follows  from 
equation 

57894  =  8.649^  +  0.2438  X  53-6^  > 
or  ^  =  2700°  Cels.* 

How  the  difference  between  the  heats  of  reaction 
under  constant  pressure  and  by  volume  is  to  be  treated 
is  well  exemplified  in  the  following  examples. 

*  More  complicated  examples  of  this  kind,  especially  those  of 
technical  importance,  will  be  found  in  Naumann's  Technisch-thermo- 
chemische  Berechnungen  zur  Heizung,  1893. 


34  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

A  substance,  any  organic  compound,  containing  C 
H,  and  O  is  combusted  completely  under  constant  at- 
mospheric pressure.  How  can  we  find,  from  the  ob- 
served heat  of  combustion  Vp,  the  difference  of  the 
intrinsic  energies  of  the  compound  and  its  combustion 
products?  In  other  words,  how  can  we  find  the  heat 
of  combustion  Vv,  at  constant  volume,  that  we  would 
observe  directly  by  the  combustion  of  the  compound, 
with  the  necessary  amount  of  oxygen  in  a  calorimetric 
bomb? 

If  the  compound  consists  of 

c  atoms  C,  //  atoms  H,  and  o  atoms  O, 
then  by  the  complete  combustion  we  would  obtain 

c  molecules  CO3  and  -  molecules  HaO, 
and  for  this  there  are  necessary 

2c  A o  atoms  =  c  -I molecules  of  o. 

'     2  42 

The  products  of  combustion  must  have  cooled  to 
the  temperature  of  the  room,  about  17°  Cels.,  before 
being  removed  from  the  calorimeter,  which  is  the  tem- 
perature at  which  the  organic  substance  and  oxygen 
were  at  first.  Therefore  for  each  molecule  of  the  sub- 
stance placed  in  the  calorimeter  c  molecules  come  out 

as   gas  (CO2)  after  the  combustion,  while  (c  +  -  —  -\ 
of  O  as  gas  went  in.     If  the  combusted  substance  it- 


THE    VOLUME  ENERGY  OF  GASES.  35 

self  is  solid  or  liquid,  then  by  the  combustion  of  one 
molecule  a  decrease  of  volume  of 

/     ,  h      o\      o       h 

c  —  {  c  -j  ----  1  =  ---  molecular  volumes 
\       4      21      2      4 

is  observed. 

If,  however,  the  substance  is  gaseous  when  placed 
in  the  bomb,  then  the  increase  of  volume  is 

-  ---  I  molecular  volumes. 
2       4 

According  to  the  law  given  on  page  29  in  the  first 
case,  which  only  will  be  treated  here, 


or  when  the  temperature  is  17°  Cels.,  or  6  =  290°,  before 
and  after  the  reaction,  then 


The  energy  equation  corresponding  would  be 
(QH.O  )  +2.  +     -  *    .0  =  < 


From  the  heat  of  combustion  of  a  substance  we  can 
find  easily  the  heat  of  formation,  for  if  we  first  produce 
a  substance  from  its  elements  and  then  combust  it,  it 
is  no  different  from  combusting  its  elements,  in  the 
proper  amounts,  directly.  If  we  start  from  the  equa- 
tions 

C  +  20  =  CO,  +  97  Cals., 
2H  +  O  =  (HaO)  +  68  Cals., 


36  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

where  the  heat  of  formation  is  somewhat  larger  than 
Thomsen  found  it  (see  page  16),  then  the  heat  of 
formation,  Bv,  of  the  above  organic  compound,  for  con- 
stant volume,  is  given  by  the  equation 

• 


The  same  heat,  under  constant  pressure,  Bp,  is  given 
by  the  equation  (see  page  31) 


or          Bp  +  VP  —  9;  rc  +  69  -  Cals. 

It  follows,  therefore,  from  the  heat  of  combustion, 
under  constant  pressure,  of  raw  sugar,  CiaHaaOn, 

Vp=  1355  Cals.; 


Julius  Thomsen  found  that  the  vapor  of  acetic  acid, 
C,H4OS,  when  at  a  temperature  of  7°  under  its  boiling- 
point,  118°,  would  burn  under  atmospheric  pressure, 
and  would  develop  227490  cals.  per  mol.  When  he 
calculated  the  molecular  heat  of  the  vapor  at  23  cals.,  he 
found  the  heats  of  combustion  at  118°  and  at  18°  equal 
respectively  to  227650  and  225350  cals.  From  this,  it 
follows,  that  the  heat  of  formation  under  constant 
pressure  is 

BP  =  97  X  2  +69  X  2  —  225  =  107  Cals., 

and  that  for  constant  volume  Bv  is  smaller  by  4  '   2  x  20 


2 


THE    VOLUME  ENERGY  OF  GASES.  37 

=  3.582  Cals.  or  105  Cals.  when  the  C2H4O2  is  liquid, 
and  smaller  by  (- 1^582  =  2. 582  Cals.  or  1 06  Cals. 

when  formed  as  a  gas  from  its  elements. 

For  ethyl  alcohol,  C2H6O,  Julius  Thomsen  round, 
under  constant  pressure,  the  heat  of  combustion  at  the 
boiling-point,  78°. 5,  to  be  341790  cals.  The  molecular 
heat  is,  according  to  Regnault,  20.8;  therefore  at  18° 
the  heat  of  combustion  VP  =  340530  cals.  In  the  ca-, 
lorimetric  bomb  at  18° 

V,  =  340530  +  (~  -  -}  582  Cals.  =  340  Cals. 

V2      4/ 

Further,  Bp  =  60,  and  B,  =  58  Cals. 

From  the  heat,  observed  by  Stohmann,  solid  stearic 
acid,  C18H36Oa,  for  constant  volume  gives  2707.1  Cals.; 
it  follows,  therefore,  that  under  constant  pressure,  at  18°, 
the  heat  is  2711.8  Cals.,  and  the  heat  of  formation  Bv  is 
263  Cals.,  and  that  of  B  is  258  Cals.,  when  we  calcu- 
late from  the  data  given  on  page  36.  Since  Stohmann 
uses  the  heat  of  combustion  of  the  diamond,  94  Cals.,  as 
that  of  carbon,  and  the  same  heat  of  combustion  for 
hydrogen  as  we  have  used,  viz.,  69  Cals.,  he  finds  Bt  = 
222.2  Cals. 

A  coal  contains  x<f>  of  hydrogen  ;  how  much  heat 
results  from  the  compression,  due  to  the  complete  com- 
bustion of  the  hydrogen  with  oxygen,  while  I  gram 
of  carbon  burns? 

Each  gram  of  hydrogen  requires  \  mol  of  O,  and 
causes  when  the  water  becomes  fluid  a  compression  of 
\  of  a  molecular  volume,  i.e.,  to  formation  of  £  .  26  = 
J  .  580  cals.  Each  gram  of  coal  causes  therefore  the 


38   PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

formation  of  —.5 —  =  1.45^  cals.,  an  amount  which 
100     4 

is  insignificant  beside  the  8000  cals.  which  one  gram  of 
carbon  develops.  Therefore  calorimetric  experiments 
on  coals  can  be  carried  on  at  constant  volume  as  well 
.as  under  constant  pressure  without  perceptibly  chang- 
ing the  results. 

In  conclusion,  it  will  be  well  to  compare  a  few 
good  measurements  of  the  heat  of  explosion  of  oxy- 
hydrogen  gas  under  constant  pressure  in  order  to  be 
able  to  judge  of  the  degree  of  accuracy  which  has  been 
reached.  Measurements  of  this  mixture  have  been 
made  more  often,  and  perhaps  more  accurately,  than 
those  of  any  other  substance.  J.  Thomsen  found  by 
three  experiments,  made  in  the  same  manner,  H2O 
being  taken  as  equal  in  molecular  weight  to  18, 

68388,     68467,     68231  cals., 

the  mean  being  68357  ca^s- 

Other  good  observers  have  found 

68433,     68924  cals. 

The  combustions  at  constant  volume  when  recalcu- 
lated to  constant  pressure  give  still  greater  results ;  for 
example,  Berthelot  found  in  this  way 

69200  cals. 

In  comparing  results  it  is  well  to  observe  which 
calorie  and  which  molecular  weight  has  been  used. 
For  H  =  1.0025,  O  =  16,  Thomsen's  mean  value  be- 
comes 68376  cals.;  for  H  =  i,  O  =  15.96,  it  becomes, 
however,  68205  cals. 


THE    VOLUME  ENERGY  OF  GASES.  39 


m 

«  ego  .2 

Compounds.  "^*      " 


g 

u 


c«~ 
<ui-i  oi 

K  K  K 

Hydrochloric  acid,  HC1  ........     220  173  .... 

Hydriodic  acid,  HI  ............    —60  192  ____ 

Water,  H20  ..................     684  ........ 

Sulphuretted  hydrogen,  H2S..  ..     (30)  46  1367 

Ammonia,  NH8  ............  .  ..      119  84  .... 

Ammonium  chloride,  NH4C1  ----     758  —39  .... 

Carbon  dioxide,  CO2  ...........     970  ....  .... 

Sulphur  trioxide,  SO3  .........    1032  392  .... 

Sulphuric  acid,  H2SO4  ..........    1929  179  .... 

Potassium  chloride,  KC1  .......    1056  —44  .... 

Sodium  chloride,  NaCl  .........     977  —12  .... 

"         hydroxide,  NaOH  .....   1019  99  ..... 

"         oxide,  Na20  ..........    1002  550  .... 

"        sulphate,  Na2SO4  ......    3286  5  ____ 

"     Na2SO4+ioH2O  3478  -188 

"         carbonate,  Na2CO3  .....    2726  56  ____ 

Zinc  sulphate,  ZnSO4  +  7H2O..   2527  —43  ____ 

Copper  sulphate,  CuSO4  -f-  H2O.   1890  93  .... 

CuSO4  ........    1826  158 

CuSO4  +  5H2O  201  1  -27 

Methane,  CH4  ............  ......     212  ----  2119 

Ethane,  C,H6  ................      274  ....  3704 

Benzene,  C6H6  ................  —137  ....  7994 

Ethylene,  C2H4  ................    -33  ....  3334 

Methyl  alcohol,  CH3OH  ........     506  ....  1822 

Ethyl  alcohol,  C2H5OH  ........      570  ____  3405 

Acetic  acid,  CHaCOOH  ........    1041  ----  2254 


40  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

The  preceding  small  table  is  condensed  from  J. 
Thomsen  and  others,*  and  will  be  found  useful  for 
practice  and  comparison. 

The  values  are  given  in  Ostwald  calories,  and  most 
are  taken  from  the  observations  and  calculations  of  J. 
Thomsen.  They  are  all  given  for  the  temperature  of 
1 8°  Cels.,  and  I  atmosphere  pressure, 

*J.  Thomsen,  Thermochemische  Untersuchungen.  Ostwald 
Lehrbuch  d.  allg.  Chem.  2  ed.,  II,  i,  1893.  (Leipzig.)  Stohmann( 
Die  Verbrennungsw&rmen  organischer  Verbindungen,  Zeit.  f.  phys. 
Chem.  6,  1890. 


PART  II. 

ENTROPY. 


CHAPTER   I. 

THE   FACTORS   OF  ENERGY. 

IN  Part  I  two  forms  of  energy  were  considered — 
that  of  heat  and  that  of  volume.  They  were,  however, 
investigated  in  different  ways.  The  energy  of  heat  was 
more  completely  treated  than  that  of  volume,  which 
appeared  only  as  a  correction  to  the  measurement  of 
heat.  Notwithstanding  this,  however,  we  are  better 
acquainted  with  the  ways  of  the  energy  of  volume  than 
with  that  of  heat.  If,  in  order  to  show  this  difference, 
we  assume  that  the  intrinsic  energy  of 'a  body  changes 
only  by  the  gain  or  loss  of  heat,  and  by  the  increase  or 
decrease  of  volume  ;  and  if  we  call  the  change,  during 
the  time  dt,  of  intrinsic  energy  dE,  and  the  increase  of 
heat  dQ,  and  that  of  work,  by  change  in  volume,  dA, 

—then 

dE  =  dQ  +  dA. 

Since  we  know,  further,  that 

dA  =  -  pdV, 

41 


42   PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

we  need  only  a  knowledge  of  dQ  to  understand  the 
equation  fully.  We  arrive  at  this  by  the  following 
analogy  between  the  different  forms  of  energy.  First, 
we  will  consider  more  exactly  the  physical  nature  of 
/  and  V. 

Let  us  imagine  a  closed  cylinder  in  which  a  piston 
AT  can  be  moved,  on  both  sides  of  which  is  a  liquid  or 
gaseous  body  which  can  increase 
or  decrease  its  intrinsic  energy  by 
changes    of   volume.       We  will 


K  assume  that  on  the  one  side  the 

FIG.  2.  ,  ^  , 

pressure  \spl  and  the  volume  is 

Vlt  while  on  the  other  it  is  /2  and  F2.  It  is  then  plain 
that  a  change  of  intrinsic  energy  can  only  occur  by  the 
decrease  or  increase  of  volume  energy  when/,  is  >.  /a. 
If  /j  is  >  /2 ,  then  the  energy  of  the  body  which  is 
under  the  greater  pressure  increases  by  —  pldVl  (de- 
creases in  reality),  and  the  other  by  —  p^dV^.  Here  dVl 
=  —  dV^,  for  the  one  volume  increases  by  the  amount 
by  which  the  other  decreases  for  the  sum  of  the  two 
volumes,  i.e.,  the  volume  of  the  cylinder  remains  un- 
changed. Further,  the  greater  pressure,^,  decreases, 
while  the  lesser, /2,  increases.  We  can  represent  it  by 
assuming  that  each  body  has  the  tendency  to  come  to 
a  lower  pressure,  and  the  greater  tendency  overcomes 
the  other. 

From  this  method  of  considering  the  question  we 
obtain  the  following  results  : 

The  first  condition  that  causes  the  intrinsic  energy 
of  a  body  to  cha'nge,  by  increase  or  decrease  of  volume 
energy,  is  the  possibility  of  a  change  in  its  volume.  If 
this  changes,  then  the  volume  of  another  body  must  of 


THE  FACTORS   OF  ENERGY.  43 

necessity  change  also,  but  the  sum  of  the  volumes 
which  are  considered  can  never  change. 

The  second  condition  that  causes  the  intrinsic  energy 
of  a  body  to  change,  by  increase  or  decrease  of  volume 
energy,  is  the  inequality  of  pressure  in  the  different 
bodies.  By  this  each  body  shows  a  tendency,  which 
increases  with  the  pressure,  to  diminish  its  pressure, 
so  that,  during  the  change,  the  greater  ones  decrease 
Avhile  the  smaller  ones  increase ;  or,  in  short,  the 
volume  energy  is  transformed  from  a  higher  to  a 
lower  pressure. 

It  is  self-evident  that  these  conditions  are  not  con- 
fined to  the  problem  just  considered  of  the  closed 
cylinder,  but  are  equally  applicable  to  all  change  of 
volume. 

The  sense  of  this  treatment  can  be  perhaps  better 
understood  in  the  form  given  by  Poincar£.  If,  in  an 
isolated  system,  i  e.,  in  a  system  in  which  only  an  ex- 
change of  energy  between  the  constituents  is  possible, 
after  any  process,  nothing  is  changed  but  the  pressure 
and  volume  of  its  two  constituents,  then  the  higher 
pressure  has  certainly  decreased,  while  the  lower  has 
increased. 

In  general,  it  is  possible  to  show  that  the  change  of 
intrinsic  energy  in  every  form  is  a  product,  IdM,  where 
/  has  the  same  meaning  as  the  pressure  in  volume 
energy,  and  J/that  of  volume,  in  volume  energy.  /  is 
called  the  intensity  of  the  form  of  energy,  and  M,  ac- 
cording to  Helm,  the  quantity  of  the  form  of  energy, 
or,  according  to  Ostwald,  the  capacity  of  the  body  for 
that  form  of  energy. 


44  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

By  the  transformation  of  the  above  considerations 
for  volume  energy  to  heat  energy,  it  is  to  be  observed 
that  the  entrance  to  the  knowledge  of  the  former  is 
won  in  an  entirely  different  manner  than  to  that  of  the 
latter.  In  the  volume  energy  we  have  the  volume  in 
tangible  form,  also  the  pressure,  and  from  the  two  we 
can  find  the  energy  ;  that  is,  from  M  and  /  we  can 


By  heat,  however,  it  is  different.  Here  by  our  organs 
of  sensation  we  have  something  that  corresponds  ex- 
actly to  pressure.  We  need  only,  in  the  above  second 
condition,  to  replace  the  word  pressure  by  tempera- 
ture, volume  energy  by  heat  energy,  to  obtain  the 
fundamental  equation  of  heat  energy.  Further,  in  the 
course  of  the  historical  development  of  the  phenomena 
of  heat,  the  theory  has  been  built  up  from  experi- 
mental facts,  so  that  here  we  are  confronted  by  the 
problem  of  how  to  find  the  function  M  from  /  and  E. 

It  is  well  to  start  from  the  fact  that  from  the  volume 
energy  A  and  the  pressure  /  we  can  find  the  function 
V\  that  is, 


But  here  there  is  a  difficulty  that  must  not  be  over- 
looked.  If  the  volume  of  a  body  increases  by  dV, 
then  the  intrinsic  energy  changes  in  a  different  man- 
ner, according  to  the  way  in  which  the  volume  in- 
creases. If  it  is  very  slow  —  as  is  the  case  when  the 
internal  pressure  is  but  slightly  greater  than  the  outer, 
and  just  enough  to  increase  the  volume  —  then  the 
body  goes  over  into  another  state  from  that  which  it 


THE  FA  CTORS   OF  ENERG  Y.  45 

would  assume  were  the  change  sudden,  as  is  the  case 
when  the  difference  of  external  and  internal  pressure 
is  large.  Then  the  single  molecules  of  the  expanding 
body  are  in  violent  motion  and  kinetic  energy  is 
liberated,  which  is  gradually  turned  into  heat  by  the 
friction.  It  is  only  after  the  disappearance  of  this  heat 
that  the  body  reaches  the  same  final  state  as  it  does 
when  the  expansion  is  gradual. 

If  we  call  the  increase  of  energy  that  takes  place  by 
the  gradual  increase  of  volume  dV,  d0E,  and  that  by  the 
sudden  increase  by  the  same  amount  dV,  dkE,  and  if 
we  call  the  kinetic  energy  developed  in  the  latter  case 
dK  then 

dkE  =  d0E  +  dK, 


dkE>dQE, 

P         P 

According  to  the  equation  above, 


, 

and 

dkE 

—  =  -dVk; 

then  -  dVk=-dV+d—-  . 

P 

Thus,  for  a  sudden  change  of  volume,  we  have 
-dV  <  -dVk, 


or  -  dVh  >        ,    d.E  <  - 

P 


46  PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 

and  only  when  dk  =  o  do  we  have 


or  -  dVh  =        , 

According  to  this,  condition  one  (page  42),  does  not 
hold  for  —  Vk  ,  i.e.,  the  sum  of  all  values  of  this  func- 
tion, that  belong  to  the  bodies  under  consideration, 
is  not  unchangeable,  but  rather  increases  steadily.  On 
the  other  hand,  —  Vkt  as  —  F,  is  a  function  fully  deter- 
mined by  the  momentary  state  of  the  body  (as  are  all 
internal  motions),  and  not  dependent  on  the  manner  of 
the  change  from  one  to  the  other. 

Thus  in  the  study  of  the  phenomena  of  heat  we 
are  in  the  position  just  described;  i.e.,  the  states  into 
which  a  body  comes,  after  absorption  of  the  same 
amount  of  heat,  are  different  according  as  the  reaction 
goes  slowly  or  rapidly.  According  to  a  method  of 
consideration  which  will  be  developed  later,  these  two 
reactions  are  distinguished  as  reversible  and  non-rever- 
sible; but  we  cannot  draw  out  the  difference  of  energy 
in  any  other  form  than  that  which  the  slow  process 
develops.  While,  in  the  above  case,  dkE  —  d0E  ap- 
pears as  kinetic  energy  of  the  particles  of  the  body 
and  can  be  measured  as  heat,  and  d0E  appears  as  vol- 
ume energy  ;  in  the  case  that  heat  is  supplied,  no  such 
distinction  is  possible. 

We  would  expect,  in  the  case  of  heat  energy,  the 
amount  of  heat,  dQ,  supplied  to  a  body  at  temperature 
0  is  comparable  to  the  volume  energy  d0E,  that  is  sup- 
plied at  pressure/,  that  is, 

dQ  ^  OdS, 


THE  FACTORS   OF  ENERGY.  47 

where  the  function  5,  comparable  to  —  FA,  is  a  quan- 
tity peculiar  to  that  state  of  the  body,  and  which  has 
the  principal  property  that  the  sum  of  all  the  terms  ^ 
can  never  decrease. 

In  order  to  prove  the  correctness  of  this  propo- 
sition, founded  on  analogy,  it  is  necessary  to  follow  a 
train  of  thought  that  was  proposed  by  Sadi  Carnot,  the 
so-called  cycle.  The  exact  study  "of  this  method,  how- 
ever, presupposes  a  knowledge  of  the  properties  of 
perfect  gases,  so,  in  our  next  chapter,  we  will  consider 
this  important  question.  — I 


CHAPTER   II. 

THE  THERMODYNAMICS   OF   PERFECT   GASES. 

LET  the  intrinsic  energy  of  I  gram  of  any  substance 
vary  by  de,  while  at  the  same  time  the  amount  of  heat 
dq  is  supplied  and  the  mechanical  work  da  is  done 
from  the  outside  (as,  e.g.,  by  the  atmospheric  pressure). 
Then,  according  to  the  principle  of  the  conservation  of 
energy,  for  every  possible  change  of  the  sort  we  have 

de  =  dq  -\-  da  ; 
and  since  da  =  —  pdv, 
(i)  de  =  dq  —  pdv, 

Here  we  must  remember  that  no  form  of  energy, 
especially  kinetic  energy,  arises  during  the  change, 
except  those  named,  i.e.,  we  assume  that  the  change 
takes  place  very  slowly.  We  will  first  assume  that  the 
amount  of  heat  dvq  is  supplied  to  the  body,  while  we 
hold  the  volume  constant,  by  which  the  absolute  tem- 
perature 0  is  increased  by  d6,  and  the  pressure  p  by 
dp ;  the  intrinsic  energy  then  increases  by 

(la)  dve  —  dvq  =  cvdO, 

where  cv  is  the  specific  heat  by  constant  volume,  that 
is,  the  heat  which  is  necessary  to  raise  i  gram  of  the 


THERMODYNAMICS   OF  PERFECT  GASES.        49 

substance  i°  Cels.  when  its  volume  is  not  allowed  to 
change.  Secondly,  we  will  assume  another  amount  of 
heat,  dQq,  to  be  supplied,  keeping  at  the  same  time  the 
temperature  constant ;  this  process  will  change,  gener- 
ally, the  volume  energy  of  the  substance,  and  so  it  will 
expand  as  well  as  change  its  internal  structure.  By 
the  Mariotte-Gay-Lussac  law,  however,  we  have  the 
equation  of  state, 

(2)  -pv  =  R6, 

and  from  this  we  see  that  the  intrinsic  energy  changes 
only  with  the  temperature — a  proposition  which  has' 
been  proven  experimentally  with  gases.  We  place, 
in  accordance  with  this,  the  change  in  the  intrinsic 
energy,  by  constant  temperature,  equal  to  zero ;  there- 
fore 

(3)  d&  =  pdv. 

If  the  two  changes,  which  we  have  considered  sepa- 
rately, follow  one  another,  then  the  intrinsic  energy,  of 
the  gram  of  substance  considered,  changes  by 

(id)        de  —  dve  +  dee  —  cvdti  =  dvq  -f-  d0q  —  pdv. 

Now  of  the  second  change ;  up  to  the  present,  we 
have  only  determined  that  it  shall  take  place  at  constant 
temperature,  or  isothermally,  but  have  not  given  the 
limits  of  the  pressure  and  volume  at  which  we  shall 
stop  the  reaction.  We  will  now  assume  that  the  pres- 
sure in  the  second  change  decreases  just  so  much  as  it 
increased  by  the  first  change.  Then  the  total  change 
which  the  gas  has  undergone  has  not  changed  its  final 


50  PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 

pressure,  and  the  total  addition  of  heat  can  be  repre- 
sented by  the  specific  heat  of  the  gas  under  constant 

pressure ;  that  is, 

(4)  dvq  +  deq  —  cpdO. 

In  connection  with  (ib)  we  have,  therefore, 

(5)  (ct-c^M=pdv. 

From  (2),  however,  by  differentiating,  we  find 
(2b)  pdv  +  vdp  =  Rde  ; 

or,  since  by  constant  pressure,  dp  =  o, 

pdv  =  RdO, 
and  from  (5) 

(6)  *,-'•  =  R. 

After  completing  the  proof  in  this  way — by  assum- 
ing that  the  difference  between  the  specific  heats  is 
unvarying,  for  substances  which  follow  the  Mariotte-' 
Gay-Lussac  law,  and  whose  intrinsic  energy  changes 
only  with  the  temperature — we  are  further  in  the  po- 
sition to  prove  the  law  by  which  the  state  of  a  gas 
changes  when  no  heat  energy,  but  only  volume  energy, 
is  applied. 

From  (ib) 

(7)  dqe  =  —  pdv  =  cvdB, 

and  from  (2)  and  (6)  it  follows  that 

(8)  pdv  +  vdp  =  (cp  -  c,)dO. 

The  elimination  of  dO  leads  to  the  relation  between 
changes  of  pressure  and  of  volume.  As  long  as  only 


THERMODYNAMICS   OF  PERFECT  GASES.         5  I 

volume  energy  is  added  to  or  removed  from  the  gas, 
then 

vdp  — -pdv  ; 

Cv 

dp   .  -Cp    dv 


We  call  this  change  of  state  —  which  takes  place 
without  any  change  in  the  heat  energy  —  adiabatic. 
For  isothermal,  changes  we  have  from  (2$),  where 
d6  =  o, 

dp      dv 


The  integral  of  this  equation,  it  is  self-evident  by 

(2),  IS 


where  />0,  v0,  and  60  mean  respectively  the  pressure, 
specific  volume,  and  temperature  in  the  initial  state  of 
the  isothermally  changed  gas.  While  we  can  thus  easily 
integrate  (10),  it  is  more  difficult  to  integrate  (9).  We 
can  only  do  so  when  we  know  more  of  the  relation  of 

the  specific  heats  —  —  x.     For   gases,  within  certain 

Cv 

limits,  this  proportion  is  constant,  i.e.,  by  equation 
(6),  the  two  specific  heats  cp  and  cv  themselves  are 
constant.  So  far  as  this,  the  integral  of  equation  (9)  is 


(12) 


52   PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

and  with  the  aid  of  (7)  and  (8)  it  follows,  further, 


yBv*-1  =  00  v*"1 ) 
(i2b)  i  i     V 

ep*-*  =  ej^-i  r 


The  proportion  ^  of  the  specific  heats  is  for  perfect 
gases  about  1.41.  Finally,  the  integral  for  (7)  is 

(13)  e  =  e,  +  c,0, 

which  shows  how  the  intrinsic  energy  of  a  gram  of  gas 
changes  with  the  temperature ;  e0  is  an  integration 
constant  peculiar  to  each  gas. 

We  must  not  forget,  however,  the  conditions  under 
which  these  equations  were  developed.  We  have 
assumed  that  there  are  in  Nature  perfect  gases  which 
have  the  following  three  properties :  (i)  They  follow 
the  Mariotte-Gay-Lussac  law ;  (2)  their  specific  heats 
are  unvarying ;  (3)  their  intrinsic  energies  .  do  not 
change  when  the  temperature  does  not. 

The  meaning  of  equations  (12)  and  (\2b)  can  be 
made  clear  by  a  simple  example.  If  a  gas  is  adiabati- 
cally  compressed  to  the  five-hundredth  part  of  its 
volume,  the  temperature  rises  from  the  initial  one  of 
17°  Cels.,  or  290°  absolute,  to  2700°,  as  the  substitution 
of  v  =  -sfov0,  00  =  290,  x  —  1.41  shows.  The  pressure 
rises  by  it  to  $oolM  or  to  6400  times  its  initial  value. 
If,  on  the  contrary,  the  pressure  is  adiabatically  raised 
to  500  times  its  initial  value,  then  the  temperature  is 
raised  to  1500°. 

For  the  following,  it  is  important  to  show  graphi- 
cally the  formulae  (n)  and  (12)  for  isothermal  and 
adiabatic  changes  of  pressure  and  volume.  Let  us 


THERMOD  YNAMICS   OF  PERFECT  GASES.        53 

imagine  in  a  cylinder,  whose  axis  has  the  direction 
v,  one  gram  of  gas.  In  the  co-ordinate  system  let  the 
volumes  be  represented  by  the  abscissae,  and  the  pres- 
sures, which  correspond  to  these  volumes,  by  the  ordi- 
nates.  Starting  from  any  initial  state,  the  point  (v,  p) 
traverses  two  curves  according  as  it  is  an  isothermal 
or  an  adiabatic.  The  curves  are  always  different,  since 
Cp,  according  to  (6),  cannot  be  equal  to  cv.  If  we  do 
not  supply  heat,  then  the  curve  followed  is  an  adiabatic  ; 
if  its  intrinsic  energy  is  unchanged,  at  the  same  time 
supplying  heat  and  volume  energy,  then,  according  to 
(13),  it  is  an  isothermal.  (Fig.  3.) 

Each  point  (z>,  /)  of  the  diagram  represents  a  possi- 


FIG.  3. 

ble  state  of  the  gas.  Conversely,  each  possible  state 
of  the  gas  is  shown  by  the  point  (v,  p\  The  tem- 
perature, by  (2),  is  also  determinable.  We  see  from 
this  that  the  change  from  any  state  P,  of  a  perfect  gas 


54  PRINCIPLES  OF  MATHEMATICAL    CHEMISTRY. 

into  any  other  state  P9  ,  can  be  shown  by  a  system  of 
curves  that  is  composed  of  the  isothermal  of  the  one 
state,  and  the  adiabatic  of  the  other.  We  have  only 
to  follow  these  curves  to  their  point  of  intersection 
P't  whose  co-ordinates  (V,  /')  satisfy  at  the  same  time 
the  equation  of  the  isothermal  through  Pl  and  the 
adiabatic  through  P9, 


and  it  follows 


- 

The  change  of  the  gas,  shown  by  the  series  of  points 
Pf  P^  ,  takes  it  from  the  one  state  into  the  other. 

Finally,  these  considerations  allow  us  to  find  the 
heat  which  is  necessary  for  this  transformation.  The 
isothermal  supply  of  heat  given  by  (3)  is 


(15)  d<q= 

by  integration  we  have 
(15*) 


The  adiabatic  supply  of  heat  is  o  ;  the  total  supply 
of  heat  is  therefore 


,„  = 


THERMODYNAMICS  OF  PERFECT  GASES.        55 

If  this  gives  the  heat  necessary  for  the  isothermal 
change  Pf  and  the  adiabatic  PP^\  then,  by  the  simple 
exchange  of  the  indices  I  and  2,  we  can  find  the  heat 
necessary  to  go  from  2  to  I  on  the  isothermal  of  2  and 
the  adiabatic  of  I,  i.e.,  through  the  series  of  points 
Pf'P.  We  obtain 

(16*)  fe  = 


CHAPTER  III. 

THE   CYCLE. 

TECHNICAL  experience  led  Sadi  Carnot  (1824)  to  the 
method — the  cycle — by  which  he  was  able  to  make 
clear  the  relation  between  heat  and  temperature. 

Saturated  vapor  that  produces  work,  in  a  steam-en- 
gine, does  it  by  going  from  the  high  temperature  and 
pressure  of  the  boiler,  to  the  low  temperature  and  pres- 
sure of  the  condenser.  The  working  body  therefore  is 
in  a  different  state  after  doing  the  work,  than  it  was 
before,  so  that,  for  the  theory  of  the  process,  we  must 
consider  not  only  the  heat  taken  and  g~iven  up  in  the 
boiler  and  condenser,  and  the  work  done  in  the  cylin- 
der, but  we  must  also  take  account  of  the  change  in 
the  intrinsic  energy  of  the  steam  itself.  The  con- 
sideration of  the  latter,  however,  can  be  neglected 
when  we  make  use  of  a  process  by  which,  as  before,  a 
transformation  of  energy  takes  place,  but  in  which  the 
working  body  is  finally  in  the  same  state  as  it  was  in 
the  beginning.  In  a  steam-engine,  this  can  be  fulfilled 
by  bringing  the  steam  and  condensed  water  back  again 
into  the  boiler  by  means  of  a  pump  (which  must  per- 
form work,  since  the  steam  is  to  be  brought  from  a 
lower  to  a  higher  pressure) ;  they  will  then  attain  the 

56 


THE   CYCLE.  $'/ 

higher  temperature  in  the  boiler.  The  volume  energy 
set  free  in  the  cylinder,  which  is  produced  at  the  cost  of 
the  pressure  and  temperature,  will  be  greater  than  that 
supplied  to  the  pump  only  by  the  volume  energy 
necessary  for  increasing  the  pressure. 

The  scheme  for  a  steam-engine  whose  steam  com- 
pletes such  a  cycle,  i.e.,  where  after  the  end  of  the 
process  the  steam  contains  the  same  energy  as  before, 
will  be 

Volume  energy  A* 

I  S 

^Cylinder j 

Lower  pressure         )  — 


I  -  Pump  < 

T 

Volume  energy  At 

Or  shorter,  according  to  C.  Neumann, 


Higher  temp.     @i  —  —  (?a     Lower  temp.     Process  A'. 

and  according  to  the  principle  of  the  conservation  of 

energy, 

(i)  a  =  0  +  a, 

i.e.,  while  from  the  energy  applied,  Q^the  amount  02 
goes  over  as  heat  to  the.  foreign  body,  and  the  amount 
Qi  —  <22  is  transformed  into  the  work  A^  —  Al  . 

The  change  of  energy  which  takes  place  during  a 
cycle  can  always  be  brought  to  the  form  of  scheme  K. 
Often  the  single  separable  parts  do  not  occur  in 


58   PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 

different  places,  as  here,  in  the  boiler  and  cylinder, 
etc.  The  complete  technical  steam-engine,  however, 
divides  the  natural  process  in  just  the  way  that  we 
must  do,  in  our  development  of  the  theory. 

We  will  now  assume  —  and  that  is  the  principal 
point  in  Carnot's  train  of  thought  —  that  the  process  can 
also  take  place  in  the  reverse  order,  i.e.,  by  reversing  all 
the  arrows  in  the  scheme  we  will  still  have  a  possible 
natural  process,  that  is, 


Higher  temp.     Qi<  --  —  <-@2     Lower  temp.    Process  (—A'). 

Here  the  heat  Q1  can  be  found  from  the  heat  Q^ 
by  the  aid  of  the  volume  energy  Q. 

When  now  another  natural  process,  that  takes  place 
between  the  same  temperatures,  is  followed  until  the 
same  amount  Q  =  Qv  —  Q^  of  heat  is  transformed 
into  volume  energy,  then  the  scheme  will  be 


Higher  temp.      (?/->  —       -  --  *Q*       Lower  temp.     Process  A". 
And  if  reversible, 


Higher  temp.     (V<—  -  -  -  <-<(V  Lower  temp.  Process  (—A"). 

If  we  follow  process  K'  by  process  (—  AT'),  we  will 
have 

t 

High  temp.   (?/-&"  --  ~  -  CV-5  Low  temp,   j  E.r°cels  i 

(  K  —  K   t 


THE   CYCLE.  59 

i.e.,  a  simple  exchange  of  heat,  equal  to 

a'-a=a'-a. 

from  a  higher  to  a  lower  temperature.     In  the  same 
way  the  arrangement  (AT  —  K^  gives  us 


High  temp.  "S-e/-*  -  -  --  ^Q*-Q*    Lowtemp.   |  5J3?  | 

which,  as  before,  is  a  simple  exchange  of  heat,  Ql  —  Q/ 
—  02  —  G/  from  a  higher  to  a  lower  temperature. 

According  to  Carnot's  theorem,  which,  on  page  44,  we 
considered  as  self-evident  :  Pure  exchanges  of  heat  can 
only  take  place  from  a  higher  to  a  lower  temperature  ;  or 
heat  has  a  tendency  to  go  from  higher  to  lower  tem- 
peratures. From  this  it  follows  that 

(2)  a'-a=o,     a~'=  a, 

when  the  process  (K  '  -  K}  is  possible,  i.e.,  when  AT  is 
reversible.  If  both  processes,  K  and  K',  are  reversi- 
ble, we  have,  further, 

a~-a~'=o,     a=  a', 

which  is  only  possible,  in  view  of  the  former  equation, 
when 

(2*)  a'  =  a 

For  the  expression  of  our  result  it  is  sufficient  that 
<22  and  <2/  are  amounts  of  heat  which  are  given  out  by 
the  same  transformation  of  energy  Q.  If,  however,  we 
bring  the  result  of  our  train  of  thought  into  the  form 

a~'>  a 

(3)  "a"  =  ¥ 


6O    PRINCIPLES  OF  MATHEMATICAL    CHEMISTRY. 

(  K  alone  ) 

according  as  \        or  '  >  is  reversible,  then  we 

(K  as  well  as  K'  } 

obtain  the  law:  The  proportion  of  the  exchanged  heat  to 
that  transformed  is  the  same  by  all  reversible  processes, 
which  take  place  between  the  same  limits  of  temperature; 
and  is  greater  by  all  non-reversible  processes  between  the 
same  limits. 

By  aid  of  (i)  it  follows  from  (3)  that 


All  reversible  processes,  that  take  place  between  the 
same  limits  of  temperature,  need  equal  supplies  of  heat  to 
transform  equal  amounts  of  heat  energy,  and  all  non- 
•reversible  ones  need  greater  amounts. 

If  we  once  determine  this  proportion  for  one  revers- 
ible process,  then  it  is  determined,  for  the  same  interval 
of  temperature,  for  all.  We  can  find  this  for  a  gas,  pro- 
vided that  the  gas  behaves  like  a  perfect  one  during  all 
the  changes  under  which  it  is  considered.  We  must 
especially  ascertain  here  that  the  proportion  is  deter- 
mined during  an  interval  of  temperature  inside  of  which 
it  acts  like  a  perfect  gas.  This  condition  is  satisfied, 
however,  by  the  two  changes  considered  on  pages  53  and 
54,  from  any  one  state  to  any  other,  by  a  Carnot  cycle. 

They  are  also  reversible  processes,  since  all  amounts 
of  heat  given  up  must  be  by  infinitely  small  differences 
of  temperature,  and  all  changes  of  volume  energy  by 
infinitely  small  differences  of  pressure,  if  the  process  is 
to  go  in  the  manner  described. 


THE    CYCLE.  6 1 

The  total  amount  of  heat,  Q,  transformed  in  the 
above  we  will  now  designate  by  q^  +  q^  in  each  cycle, 
and  place  it  equal  to  Q.  Then 

(4)  *»  +  ?«  =  G1=i2. 

The  heat  q^t  applied  at  the  higher  temperature, 
that  we  have  called  Q19  we  will  now  designate  by  Qlt 
and  that  amount  at  the  lower  temperature,  £21,  formerly 
<22,  we  will  term  Qt. 

(5)       *,.  =  0,  =  a;  4»  =  Q,  =  -Q,- 

The  equation  on  page  55,  (17),  then  takes  the  form 

Q-Q.  |  g._0     0.  |  Q-g.-0. 

-      +~  '     "+"~ 


from  which  it  follows 


-e.      0.      c,      *, 

From  (3),  then,  we  have  for  all  general  processes 


TO^  rqtio  of  the  transformed  heat,  to  that  given  off 
at  the  lower  temperature  is,  at  greatest,  equal  to  the 
ratio  of  the  difference  of  temperature,  to  the  lower  tem- 
perature. 

Further,  we  have 


62   PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

The  ratio  of  the  transformed  heat  to  that  absorbed 
is,  at  greatest,  equal  to  the  ratio  of  the  difference  of 
temperature,  to  the  temperature  at  which  it  is  taken  up. 

For  example,  if  a  steam-engine  works  with  a  boiler- 
pressure  of  six  atmospheres  and  a  condenser-pressure 
of  one-half  atmosphere,  then  the  temperature  at  which 
the  heat  is  absorbed,  according  to  the  table  for  the 
tension  of  steam,  is  159°  Cels.  or  432°  absolute,  and  the 
temperature  at  which  the  heat  is  given  up  is  46°  Cels. 
or  319°  absolute:  the  difference  of  temperature  is  then 
113°.  Therefore,  no  machine  with  such  a  difference  of 
temperature  can  transform  more  than  113/319  of  the 
heat  given  up  in  the  condenser  into  work,  or  more 
than  113/432  of  that  absorbed  by  the  boiler. 

With  the  assistance  of  (i),  it  follows  from  (7)  that 

(8)  +±ofor»—  gfcydes.      B 


This  relation  holds  for  every  cycle,  by  which  the 
amounts  of  heat  Q^  and  <2U  have  been  absorbed  at  the 
temperature  ^  and  0a  respectively,  if  no  other  change 
of  heat  has  taken  place.  From  the  conception  of  the 
cycle  it  follows  that  the  total  amount  of  heat  ab- 
sorbed, Q1  -f-  <22  —  Q,  is  equal  to  the  amount  of  the 
other  energy  which  is  given  off,  no  matter  which,  6^  or 
03,  is  the  higher  temperature. 

The  case  of  a  body  going  from  a  state  I  to  another 
state  2,  in  a  way  not  consisting  of  adiabatics  or  isother- 
mals,  can  now  be  made  clear.  For  this  purpose  it  is 
necessary  to  complete  the  change  of  state,  by  an 
isothermal  going  through  I  and  an  adiabatic  going 


THE    CYCLE. 


through  2,  to  a  cycle  i2Pi,  and  to  cause  the  parts  2Pf 
and  Pi  to  take  place  reversibly,  so  that  the  whole  proc- 
ess is  reversible  or  not  according  as  12  is  or  not  (Fig.  4). 
Then  we  must  draw  through  the  separate  points  of  the 
line  12  adiabatics,  e.g.,  through  the  points  a,  b,  c  the 
adiabatics  aa',  bb',  cc' .  In  this  way  we  divide  the 
cycle  into  many  single  ones,  as  abb'a'a,  bcc'b'b,  each  of 
p 


FIG.  4. 

which  will  be  the  more  exactly  fulfilled  by  Carnot's 
cycle,  the  nearer  the  points  a,  b,  c,  etc.,  are  together. 
For  example,  abb' a1  a  lies  between  the  Carnot  cycles 
amb'a'a  and  nbb'a'n  when  am  and  nb'  are  isothermals. 
These  Carnot  cycles  are  in  general  reversible  if  the 
process  12  is  reversible. 

If  we  call  <212  the  heat  that  is  supplied  in  the  trans- 
formation 12,  and  <2*]2  that  supplied  for  reversible 
Carnot  cycles,  i.e.,  those  consisting  of  adiabatics  and 
isothermals,  then  the  heat  belonging  to  one  division 
ab  of  the  line  12  is  dQ^.  and  that  to  the  division  b'a'  is 
dQ*n  or  —  dQ*^.  Since  now  dQ^  is  enclosed  between 


64  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

the  amounts  of  heat  which  correspond  to  the  isother- 
mals  am  and  tib,  then  for  each  process,  as  abb'  a'  a,  if  it 
is  a  reversible  process,  according  to  (8),  we  have 


_  0 


and  for  all  single  processes,  or  for  the  total  process 

12P'  I, 


according  as  it  is  a  non-reversible  or  a  reversible  proc- 
ess, by  which  formulae  the  integral  is  to  be  taken  in  the 
way  that  the  state  I  is  transformed  into  2.  Whichever 
way  we  use,  for  the  change  of  state  (i,  2),  the  expression 


will  always  have  the  same  value  for  reversible  processes 
(as  by  the  Carnot  method),  and  smaller  values  for 
every  other  that  is  not  reversible. 

This  fact  leads  us  to  the  simplest  mathematical  ex- 
pression of  the  formula,  in  that  we  make  use  of  the 
function  5,  for  which  each  body,  in  a  certain  state,  has 
a  certain  value,  according  as  to  what  arbitrary  value  we 
give  to  S  for  any  standard  state  of  the  body.  If  we 
give,  for  example,  a  point  in  the  field  (v,p)  of  a  possible 
state,  designated  by  O,  the  value  50,  and  the  points  i, 
2,  3,  etc.,  the  values  St  St,  .  .  .  which  are  given  by  the 
equations 


THE   CYCLE.  65 

then 


for  reversible  changes  ;  or  in  general, 


Finally,  for  an  infinitely  small  change  of  state  we  have 
(ii)  dQ<SdS. 

This  function  5  is,  as  the  intrinsic  energy,  deter- 
minable  only  to  an  arbitrary  constant  S0.  It  was  called 
by  Clausius  the  entropy.  We  can  prove  by  this 
our  assumption  (page  46)  that  5  has  the  property 
by  which  it,  as  well  as  —  Vk  ,  can  never  decrease. 
We  imagine  every  possible  change  of  energy  divided 
into  a  transformation  of  energy,  that  takes  place  in  a 
body,  and  a  transmission  of  energy,  unchanged,  from 
one  body  to  another:  then  it  is  easy  to  see  that  by 
each  change  of  energy,  in  which  heat  comes  into  play, 
we  will  have  an  increase  of  entropy  for  the  body  which 
absorbs  heat,  and  a  decrease  for  the  one  which  gives  it  up. 
That  the  first  is  in  the  preponderance  is  shown  by  (11), 
from  Carnot's  principle,  which  gives  heat  the  tendency 
to  go  from  higher  to  lower  temperatures.  For  if  a 
body  absorbs  the  heat 

dQ'<  6dS, 

another  must  give  it  up.     For  this  absorption  of  heat 
however,  the  equation 

dQf<0'dS' 
holds,  in  which          dQ  =  —  dQ. 


66  PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 

If  by  dZ  we  understand  the  increase  of  entropy 
that  takes  place  by  the  transformation  of  heat, 

dZ=dS+dSf, 
it  follows  that 


According  to  the  law  of  intensity,  0'=?  ^>  therefore, 
dZ  can  never  be  negative. 

(12) 


If  a  system  has  completed  a  cycle,  its  entropy,  by 
(10),  need  not  increase.  Since,  in  the  interior  of  the 
system,  the  transmission  of  heat  cannot  lead  to  a  de- 
crease of  the  entropy  (12),  but  is  open  to  an  increase, 
the  entropy  taken  up  from  the  outside  during  the  cycle 
can  well  be  negative,  but  never  positive  (Poincare). 

It  is  worthy  of  notice,  further,  that  when  an  adia- 
batic  takes  place  reve«rsibly,  the  entropy  does  not 
change,  as  is  shown  by 

dQ  =  BdS 
when  dQ  =  O. 


CHAPTER  IV. 

THE  ENTROPY   OF  GASES  AND  GAS    MIXTURES. 

THE  function  called  entropy  is  known  to  us  for  per- 
fect gases.  The  formula  (16),  on  page  54,  gives  the 
heat  which  is  necessary  for  a  Carnot  reversible  change 
of  state  for  i  gram  of  gas.  This  function,  which  is,  as 
before  mentioned,  determinable  only  to  a  constant 
(page  65),  is 

(I)  ,,-^^+,/J 

where  sl  and  s^  are  the  values  of  s  for  I  gram  of  gas 
in  the  states  I  and  2.    The  function  (s)  is  therefore 


(2a)  *V5£-+S* 

p  v 

where  s0  is  the  arbitrary  constant.    From  the  equation 
of  state  for  a  gram  of  gas 

(3)  pv  —  R6.          where  R  =  cp  —  cv  ,     (3^) 

we  have 

(2b)       s  =  s0  +  Rlv  +  cvl(6R)  =  j/  +  Rlv  +  cjd 
(2c)        S  =  s0+Rfy  +  cpl(6R)  =  s."  -  Rip  +  cPW, 

where  s  '  and  s9"  are  again  constants. 

67 


68   PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY, 

The  entropy  of  M  grams  of  gas  can  be  found  by 
multiplying  (20),  (2b),  and  (2c)  by  M\  and  that  for 
I  mol  by  multiplying  by  m,  when  m  is  the  molecular 
weight  of  the  gas. 

Let  us  now  imagine  in  the  volume  F,  at  the  tem- 
perature •&,  or  the  absolute  temperature  0  =  273  -f  •&, 
a  number  of  different  gaseous  bodies,  which  can  be 
regarded  as  perfect  gases  ;  and  let  us  further  assume 
that  we  have  the  masses,  of  the  single  chemically  dif- 
ferent gases,  equal  to 

(4)  M1  =  n1m1)      M,  =  njnt,     ..., 

where  ml  ,  wa  ,  ...  are  the  molecular  weights  of  the 
different  gases,  and  »,,«„,  ...  the  number  of  mols 
of  each  gas  present.  Further,  that  RltR3t  ...  are 
the  gas  constants,  and  the  terms  /,  ,  /a,  /3  ,  .  .  .  of  the 
formulae 

(5)  A  V  =  njnJR.fl,     p*V  =  njn&B,  .  .  . 

are  the  partial  pressures  of  the  gaseous  constituents. 
From  Avogadro's  Law,  pages  25  and  26,  we  have 

(ejm.R^m^  =  .  .  .  =  R0  =  84800  g.*cm.  :9C.=2  cals. 

From  (5)  with  the  aid  of  (6)  we  then  have 
(7) 
where 


(85)  N  =  «,  +  »,  +  .  .  . 

Since  N  is  the  total  number  of  mols  in  the  mixture, 
NR0is  the  gas  constant  for  the  same,  and  P  is  the  total 
pressure. 


ENTROPY  OF  GASES    AND    GAS  MIXTURES.       69 

We  divide  (5)  by  (7)  and  obtain 
A  _  »i  _          A  _  ».  _ 


where  £,  £,,  ...  are  the   concentrations*  of  the  single 
constituents  of  the  mixture.     It  is  to  be  observed  from 
(8)  and  (9)  that 
(10)  Cl+C,  +  ...=  i. 

The  specific  volumes  of  the  constituents  are 
V_R1B_RQ8_     R06  Ro6 


and  the  volume  of  each  mol  of  the  single  members  is 
RQ0    i  Rfi    i 

m^-'--p^;  "rt^'-p-c?- 

With  the  help  of  this  formula  it  is  possible  to  bring 
the  expression  for  the  entropy  of  the  constituents  into 
a  different  form.  The  entropy  ^  of  the  unit  of  mass 
of  the  first  constituent  is  then,  by  (26)  and  (11), 


and  this  formula  is,  by  (6)  and  (3^),  transformed  into 
(13)       *,  =  [>OI  +  c  ,/KJ  +  cflt0  -  RJP  -  RJC, 


One  mol  of  the  first  constituent  possesses,  then,  the 
entropy 

(13$)  //v,  =  [^^0,+  ^0,/^J  +  M&J0  ~  RJP-  RJC, 
=  mlS^  m^lR.O  -  R0IP-  RJC,. 

*  Plank,  Weid.  Ann.  32,  1887;  see  also  the  reference  on  page  ir. 


7O  PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 

The  expression  in  the  bracket  can  be  regarded  as  a 
new  constant,  as  was  the  quantity  s0  earlier. 
From  (2b)  and  (n)  it  follows,  further, 


In  all  cases  in  which  the  entropy  of  the  whole  system 
is  the  sum  of  the  entropies  of  its  constituents,  we  ob- 
tain from  (13$)  and  (9)  the  total  entropy.  Thus 

•S  =  S0  +  \njnjfr  +  njnj  a  +  .  .  .  ]/0 
-tf.lX/A+w./A  +  ...]. 

(15)       s  =  s.  +  cie-Rj(p*,pr...). 

It  is  self-evident  that  the  total  entropy  is  the  sum 
of  the  entropies  of  the  constituents,  only  when  the 
mixing  is  a  reversible  process.  The  simplest  case  of 
application,  i.e.,  when  all  constituents  are  alike  and 
present  in  the  same  amounts,  in  (15),  leads  to  direct 
contradiction  to  (2c),  as  Gibbs  has  already  observed  and 
C.  Neumann*  lately  reiterated. 

If  we  assume  k  constituents,  then 


=».=-=i* 


and  from  (9) 

j 
and  from  (15) 


Leipzig,  Ber.,  1891. 


ENTROPY  OF  GASES  AND  GAS  MIXTURES.  Jl 
while  from  (2c)  we  obtain 

S=S0  +  cpl0  -  R.NIP. 

We  see  from  this  that  this  method  of  consideration, 
that  each  of  the  like  particles  fill  the  volume  V  with 

p 

the  partial  pressure  —  ,  is  apparently  wrong.     But  we 

A? 

can  account  for  it,  for  here  we  have  a  mixture  of  like 
particles  which  is  not  reversible,  as  a  mixture  of  unlike 
ones  would  be,  and  therefore  the  conditions  of  devel- 
opment are  not  fulfilled. 

If,  on  the  other  hand,  we  imagine  the  mixture  of 
like  particles  to  be  made  in  the  usual  way,  so  that  all 
parts  are  under  the  common  pressure  P,  and  each  fills 
the  /&th  part  of  the  volume  Fi  then  the  process  is  a  re- 
versible one. 


CHAPTER  V. 

THE  RELATIONS  BETWEEN  HEAT  AND  VOLUME 
ENERGY. 

THE  principal  mathematical  result  of  the  investiga- 
tion begun  on  page  41  is  a  more  complete  method  of 
expression  of  the  principle  of  the  conservation  of  energy. 

(1)  dE  =  dQ  +  dA. 

This  equation  indicates  that  a  body,  during  the  time 
dt,  has  had  (positive  or  negative)  heat  or  work  (dQ  and 
dA)  applied  to  it ;  and  that  their  sum  is  the  increase  of 
intrinsic  energy.  If  the  work  dA  is  done  only  by  the 
change  of  volume,  under  the  influence  of  the  outer 
pressure  (e.g.,  of  the  atmosphere)  />,  then 

(2)  dA=-pdV. 

We  know  further,  however,  from  a  previous  chapter, 
that 

(3)  dQ<6dS, 

where  6  is  the  absolute  temperature  and  5  the  entropy, 
and  we  also  know  that  5  is  a  function  of  a  body,  just  as 
its  volume  is,  and  that  is  known. 

From  this  the  principle  of  the  conservation  of 
energy  assumes  the  form 

(4)  dE<6dS-pdV. 

72 


HEAT  AND    VOLUME  ENERGY.  73 

It  is  important  here  to  keep  the  fact  clear  that  dE, 
dS,  and  dV  represent  very  various  changes,  namely,  all 
those  infinitely  small  changes  that  are  possible  under 
the  condition,  that  only  gain  or  loss  of  heat  and  of 
volume  energy  can  take  place. 

We  could,  for  example,  let  S,  F,  and  E  represent 
co-ordinates,  so  that  all  states  (that  could  occur  by  re- 
versible processes,  i.e.,  those  to  which  the  equation 

(5)  dE  =  6dS-pdV 

would  correspond)  can  be  regarded  as  points  on  a  sur- 
face, whose  tangential  planes  are  determined,  in  each 
point  given,  by  the  co-ordinates  S,  F,  and  E,  by  the  aid 
of  6  and  /.  In  this  way  we  can  readily  understand 
that  from  any  state  there  aie  many  changes  possible; 
for  example,  one  by  dS  =  o,  but  dV  not  equal  to  o ; 
another  by  dS  =  dV,  etc. 

Among  other  things,  equation  (4)  shows  us  that  the 
entropy  can  never  decrease  by  any  change  by  which 
the  volume  energy  and  intrinsic  energy  remain  un- 
changed, for  from  dE  =  o,  dV  =  o,  it  follows  that 
o  <JS. 

The  laws  which  concern  constant  volume  are  similar 
to  those  for  constant  temperature  or  oressure.  We 
obtain  them  by  transformation  of  (4) : 

dE  <  d(6S)  -  SdV  -  pdV. 

(6)  d(E-  0S)  ^  —  Sd6-  pdV. 

dE 5 d(8S)  -d(pV)-  Sdd  +  Vd£. 
-  SdO 


74   PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 

The  function  E  +/F  has  already  been  introduced 
on  page  30. 

Equation  (6)  shows  us  that  by  isothermal  processes, 
by  which  the  volume  is  not  changed,  the  function 
E  —  QS  cannot  increase ;  while  (7)  shows  that  in  iso- 
thermal processes,  by  which  the  pressure  cannot  in- 
crease, another  function,  E  —  SS  -\- p  F,  cannot  increase. 
The  function  E  —  OS  is  called  the  tJiermodynamical 
potential  by  constant  volume,  or  also  t\\e  free  energy  by 
constant  temperature.  Corresponding  to  this,  E  —  6S 
-\- pV  is  called  the  tJiermodynamical  potential  by  con- 
stant pressure,  or  the  free  energy  by  constant  pressure 
and  temperature. 

The  usefulness  of  these  functions  for  the  mathemati- 
cal treatment  of  chemical  processes  has  been  shown  by 
the  numerous  and  careful  investigations  of  Duhem.* 

The  much-combated  so-called  "  third  principle  of 
thermodynamics  "  of  Berthelotf  (the  two  others  express 
the  laws  of  energy),  according  to  which  a  chemical  re- 
action strives  to  form  substances  by  which  the  most 
heat  is  developed,  has  theoretically  no  foundation,  and 
has  shown  itself  to  be  of  no  practical  value.  The 
laws  just  derived  must  take  its  place,  especially  the  law 
which  is  so  similar,  dS  ^  o.  This  law  has,  however, 
already  been  applied  to  chemical  processes  by  Horst- 
mann. 

We  will  apply  equation  (4),  as  we  did  in  a  previous 

*  Duhem,  Le  potential  thermodynamique  (Paris,  1886) ;  and  since 
in  numerous  articles,  especially  in  the  Traveaux  et  memoires  des 
facuhes  de  Lille,  1-5,  III,  n,  12,  13. 

f  Essai  de  m6canique  Chimique  (Paris,  1879). 


HEAT  AND    VOLUME  ENERGY.  ?$ 

chapter,  to  the  case  that  the  substance  considered  is 
present  in  two  different  states ;  and  that  the  amount  of 
substance  in  the  one  state  can,  by  transformation  from 
the  other,  be  increased,  and  vice  versa.  For  example, 
formula  (4),  applied  to  a  mixture  of  water  and  ice,  will 
express  the  changes  of  energy  for  the  transformation  of 
a  small  amount  of  water  into  the  solid  state,  or,  con- 
versely, those  by  the  melting  of  a  small  amount  of  ice 
to  water.  It  will  be  well  now  to  develop  this  formula 
(4),  for  the  case  of  a  system  where  the  constituents 
are  in  the  state  of  exchange,  from  the  formulae  of  the 
two  single  constituents. 

We  make  use  of  the  name  as  proposed  by  Gibbs  of 
phases  to  distinguish  the  states  in  which  the  substance 
is  present. 

The  single  homogeneous  bodies  of  a  system  we 
will  call  their  phases,  in  so  far  as  their  possible  changes 
consist  of  an  exchange  of  substance ;  or  in  so  far  as  not 
to  regard  the  quantity  and  form  of  these  bodies.  Ac- 
cording to  this,  water  and  ice  are  to  be  considered 
as  two  phases  of  the  same  substance.  Accordingly,  we 
will  designate  two  bodies  as  phases  of  one  and  the 
same  substance  when,  to  their  possible  changes,  the  sum 
of  the  increase  of  mass  is  equal  to  zero,  i.e.,  the  one 
increases  by  just  so  much  as  the  other  decreases. 
Therefore,  states  of  aggregation,  states  of  dissociation, 
allotropic  and  isomeric  modifications,  are  to  be  looked 
upon  as  phases  of  one  and  the  same  substance. 

Let  us  now  assume  that  the  intrinsic  energies  of  two 
phases  of  the  same  substance,  for  whose  changes  we 
wish  to  find  the  formulae,  are  E^  and  E^ ;  that  the  en- 


76  PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 

tropics  are  Sl  and  52;  the  volumes  Vl  and  F2;  the 
masses  M^  and  M^  ;  the  temperatures  #,  and  #2  ;  and  the 
pressures/,  and  /2.*  Each  of  the  phases  (water  and  ice) 
is  a  body  that  can  absorb  or  give  up  the  heat  energy 
6dS  and  the  volume  energy  —  pdVy  but  each  is  capable 
of  another  change.  Its  energy  can  increase  in  such  a 
way  that  its  mass  increases.  Other  changes  of  energy. 
as  electrical,  etc.,  we  will  not  consider  here. 
We  have 


dE^  BldSl  -  p,dV,  +  n,  dM,  , 

< 


where  77,  =  —     ,  or  the  increase  of  energy  by  the  re- 


versible increase  of  the  unit  of  mass,  when  it  takes  place 
without  change  of  heat  or  volume  energy.  These 
equations  would  also  hold  if  the  two  phases  were  sepa- 
rated, so  that  no  exchange  of  energy  could  take  place. 
We  will,  however,  turn  our  attention  now  to  changes 
that  take  place  during  an  exchange  of  energy.  Among 
all  the  possible  changes  for  which  the  above  equations 
hold,  there  are  also  such  by  which  no  supply  of  energy 
from  outside  and  no  loss  of  energy  from  inside,  i.e.,  no 
internal  change  of  energy,  takes  place  ;  but  the  system 
is  isolated,  as  in  the  case  of  equilibrium.  For  such  sys- 
tems we  have  the  condition 

(9)  d 


*  These  quantities  are  not  completely  independent  of  one  another, 
e.g.,  the  specific  gravity  gives  a  relation  between  M,  V,  0,  and/. 


HEAT  AND    VOLUME  ENERGY.  /7 

that  is,  the  amount  of  energy  of  the  system  suffers 
no  change.  Then  only  an  exchange  from  one  to 
the  other  of  the  phases  is  possible.  Further,  for  the 
internal  changes,  we  have  the  condition 

(10)  dMl+dM,  =  0,     dM^-dM,-, 

that  is,  that  the  mass  of  the  system  is  constant. 
Further, 


for  a  change  in  the  volume  rilled  by  the  system  is  only 
possible  by  increase  or  decrease  of  volume  energy.  In 
a  corresponding  way  it  follows,  finally,  when  all  ex- 
change of  heat  between  two  phases  is  reversible,  that 

(12)  dSl  +  dS,  =  o,     dS,  =  -  dSiy 

because  reversible  entropy  changes  of  the  system  have 
the  same  meaning  as  increase  or  decrease  of  heat. 
According  to  experience,  it  is  only  possible  for  a  system 
to  have  a  reversible  exchange  of  heat  when  the  tem- 
perature and  pressure  have  become  equalized.  By  (8), 
by  addition,  it  follows 

(13)  o  <  (0,  -  0,)dSt  -  (A  -p^dV,  +  (tf,  -  njMf, 

When    pressure    and    temperature    have    become 
equalized,  then 

(14)  ^  =  0,,    A=A 
or 


(15)  o  <o  .  dS,  -  odV  +  (71,  - 

Since   now  a   possible   change  of   a  system  is   an 


78  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

increase   or  decrease    of   the   mass   of   a   phase,  then 
dM,  ^o,  and  from  (15) 

(16)  n,  =  772. 

This  holds  with  the  exception  of  the  case  when 
one  of  the  two  phases  does  not  exist,  but  whose  for- 
mation is  possible.  For  then  only  an  increase  of  mass 
is  possible,  or  dM^  >  o ;  and  so  from  (15)  we  have 

(16*)  n^n,. 

The  function  II,  whose  absolute  mass,  according  to 
(8),  is  the  square  of  a  velocity  (cm.  :  sec.)2,  obtains  from 
this  very  great  importance ;  it  is  called  by  Gibbs  the 
potential  of  the  substance.  Since  later  the  word  poten- 
tial is  used  in  another  sense,  we  will  call  H  the  chemical 
intensity*  of  the  substance,  in  keeping  with  the  analogy 
between  77"  and/  and  6.  As  the  pressure  and  tempera- 
ture, so  the  chemical  intensity  of  phases  in  contact 
becomes  equalized. 

We  can  now  express  our  results  in  the  following 
manner  :  When,  after  entrance  of  equilibrium  of  tem- 
perature and  pressure  in  an  isolated  system,  consisting 
of  two  phases  of  the  same  substance,  changes  (of  tem- 
perature and  pressure)  are  still  possible,  by  which  one 
or  the  other  of  the  phases  increase,  then  both  phases 
have  same  chemical  intensity,  if,  however,  only  one 
phase  is  present,  then  the  other  can  only  appear  when 
it  has  not  a  smaller  chemical  intensity  than  the  other. 

*  This  term  is  only  proposed  as  a  temporary  name.  By  deeper 
mathematical  knowledge  of  chemical  phenomena  77  will  undoubtedly 
be  called  by  its  old  and  proper  name,  affinity. 


HEAT  AND    VOLU^^^I^KG^^          79 


In  conclusion,  it  is  easy  from  (8)  to  deduce  (4), 
which  holds  for  the  system  as  a  whole.  Equation  (4) 
holds  for  the  changes  of  a  system  that  consists  of  the 
change  of  the  total  heat  or  volume  energy.  Increase 
or  decrease  of  mass  are  alone  excepted,  therefore 

(17)  dM,  +  dMt=:0. 

If  the  exchange  of  mass  in  the  system  is  reversible, 
then 

(18)  71,  =  77,. 

Temperature  and  pressure  are  also  equalized,  or 

(19)  0,  =  ft,  =  0,  A=A=/- 

Then  follows,  by  addition  of  (8), 
dE<  6dS  —  pdV, 

where  E,  5,  and  V  are  the  energy,  entropy,  and  vol- 
ume of  the  entire  system,  and  the  lower  sign  (  =  )  holds 
when  the  total  heat  and  volume  energy  only  undergo 
a  reversible  change. 

a.  Isothermal  Change  in  a  System  of  Two  Phases 
of  the  Same  Substance. 

We  will  now  find  the  heat  which  is  necessary  to 
transform  one  phase,  into  another  of  the  same  sub- 
stance, which  is  in  contact  with  it,  when  during  the 
transformation  neither  the  temperature  nor  the  pressure  • 
changes,  and  the  process  is  reversible. 

We  will  call  a  transformation  reversible  when  the 
supply  of  a  certain  amount  of  heat  transforms  just  as 


80  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

great  an  amount  of  phase  I  into  phase  2  as  will  be 
formed  of  phase  I  from  phase  2  by  the  removal  of  the 
same  amount  of  heat. 

In  order  to  bring  the  constancy  of  temperature  to 
a  simple  expression,  we  turn  to  equation  (6)  on  page 
73,  where  we  have  the  differential  of  the  temperature. 
We  have,  therefore,  for  a  reversible  change, 

(1)  -  d(E  -  OS)  =  SdO  -\-pdV. 

If  we  now  consider  6  and  V  as  independent  vari- 
ables, upon  which  all  the  others  in  the  equation  depend, 
as  on  page  73  (where  they  were  the  co-ordinates  of  a 
point,  so  that  the  value  of  the  thermodynamical  poten- 
tial (E  —  OS)  is  represented  by  a  point  on  a  curved 
surface),  then  .Sand/  are  negative  partial  differentials 
of  this  function  with  respect  to  0  and  v  respectively. 
Or 

(2)  95  =  g# 
-  30 


Now  we  can  calculate  the  heat  dQ,  which  is  added 
to  the  intrinsic  energy  of  the  system  by  an  infinitely 
small  change  of  state.  If  0  and  v  are  increased  by  an 
infinitely  small  amount,  then  the  intrinsic  energy  will 
be  increased  by 

(3).  dE  =  6dS  -  pdV  =  dQ  +  dA, 

and  the  energy  which  appears  as  heat  will  be 


4) 


HEAT  AND    VOLUME  ENERGY.  8  1 

or   when   the   process   takes   place   isothermally,    i.e., 
dB  =  o, 

(5)  dQ  =  edS  = 

or  from  (2) 

(6)  dQ  =  OdS  =  e^dV. 


Since,  however,  it  was  provided  that  during  the 
process  the  pressure  does  not  change  (as  if,  e.g.,  the 
process  takes  place  under  atmospheric  pressure),  then  p 
is  not  dependent  upon  V  \  no  matter  how  much  heat  is 
supplied  isothermatically,  or  how  much  of  one  phase  is 
transformed  into  the  other,  the  pressure  remains  the 
same.  The  pressure,  however,  can  depend  upon  the 
temperature,  i.e  ,  an  isothermal  process  by  another 
temperature  is  only  possible  under  another  pressure. 

From  these  considerations  we  can  go  from  infinitely 
small  changes  of  volume  and  heat  directly  over  to  finite 
ones.  We  then  have 


(7) 


i.e.,  in  order  to  bring  about  a  volume  increase  AV>  in  a 
system  of  two  phases  of  the  same  substance,  by  the 
reversible  transformation  of  one  phase  into  the  other, 
by  constant  pressure  and  temperature,  we  must  supply 
the  amount  of  heat  equal  to  AQ.  This  amount  of 
heat  is  proportional  to  the  change  of  volume  caused 
by  it,  as  well  as  to  the  absolute  temperature  of  the 

process,  and  finally  to  the  quantity  ~,  which  depends 


82   PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

upon  the  influence  of  the  pressure^*  upon  the  tempera- 
ture 6  of  the  transformation. 

This  law  was  developed  essentially  by  Clapeyron 
in  1834,  and  can  be  regarded  as  the  most  important 
result  of  the  mechanical  theory  of  heat  in  its  old  form 
of  development. 

The  increase  of  intrinsic  energy  during  the  volume 
increase  A  V  follows  from  (3)  and  (7)  as 


(8) 


Particularly  important  for  the  knowledge  of  the 
amounts  of  heat  AQ  is  the  quantity  ^,  a  differential 

that  we  can  understand  better  when  /,  as  a  function 
of  6,  is  graphically  produced. 

We  can  readily  see  then  that  for  each  temperature 
6  a  value  of  the  differential  corresponds.  The  curve 
which  shows  the  relation  between  /  and  6  is  called  the 
tension  curve  of  the  system  of  two  phases  considered. 


b.  Changes  in  the  State  of  Aggregation. 

The  melting  of  ice  and  the  freezing  of  water  are 
reversible  processes  to  which  we  can  apply  our  general 
results.  Since  ice,  by  melting,  decreases  its  specific 
volume  or  increases  its  specific  weight,  AQ  and  AV  in 

this  case  have  different  signs;  therefore  ~  is  negative, 

and    the    melting-point   sinks   by  increasing  pressure. 
This   consequence   was   found,    from   the   mechanical 


HEAT  AND    VOLUME  ENERGY.  83 

theory  of  heat,  by  J.  Thomsen  in   1850,  and  proven 
experimentally  later  by  W.  Thomson  and  others. 

To  show  how  the  units  of  mass  are  to  be  treated, 

we  will  now  find  the  value  of  the  differential  ^     If 

we  express  the  energy  by  g.*cm.,  then   I  gram  of  ice 
takes  up  79.87  cals.  of  heat-energy,  or 

AQ  =  79.87  X  43250  g.*cm. 

The  specific  gravity  of  ice  is  0.918,  the  specific 
volume  therefore  1.09,  while  the  specific  volume  of 
water  at  o°  does  not  differ  from  I  except  in  the  fourth 
decimal  place.  Therefore  we  must  substitute,  for 
AV,  —  0.09  cm.3,  and  it  follows  thst 

dp       79.87  X  43250  g.* 


9#    "  273  -(—0.09) 

When  the  pressure  increases  by  140  kg.*cm.2.,  the 
melting-point  is  lowered  i°,  or  one  atmosphere  increase 
of  pressure  lowers  the  melting-temperature  by  0.0074° 
Cels. 

From  the  heats  of  solution  for  I  gram  of  solid  and 
fluid  acetic  acid,  —  40.80  and  -f-  5-58  cals.  respectively, 
we  find  the  heat  of  melting  to  be  46.42  cals.  The 
increase  of  volume  by  the  melting  of  one  gram  is 
0.1595  cm.a  ;  therefore  we  have  (Visser) 

A  V 


AQ  ~  a.  -     43250    '   g' 

Since  the  melting-point  is  16.6°, 

d9  '    (273  +  16.6)0.003436  0.0238         "  C- 

dp  4325°  atmospheres 


84  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

while  from  observation  the  increase  of  temperature  by 
an  increase  of  pressure  of  one  atmosphere  is  0.02435° 
Cels. 

Evaporation  is  also  a  reversible  isothermal  process 
and  so  must  be  considered  here.  Since  the  tensions 
corresponding  to  the  different  boiling-temperatures  are 
easy  to  observe  (i.e.,/  is  easily  found  experimentally 
from  0),  equation  (7)  will  prove  valuable  for  the  deter- 
mination of  the  increase  of  the  specific  volume  by 
evaporation. 

From  the  table  of  Zeuner,  derived  from  Reginault's 

observations,  the   function  —n  has,  for  water  at    ico° 

oV 

Cels.  or  0  =  373°,  the  value  2.72,  expressed  in  centi- 
meters of  a  column  of  mercury  ;  therefore 


Since,  further,  the  heat  of  evaporation  of  one  gram 
of  water  at  100°  Cels.  is,  according  to  Regnault,  536.5 
cals.,  with  Joule's  equivalent  of  heat  we  have 


This  calculation  is  in  Zeuner's  table  for  water  from 
—  20°  to  200°  Cels.,  and  also  for  a  number  of  other 
liquids,  investigated  by  Regnault.  If  we  calculate  the 
specific  volume  of  steam  at  100°,  under  the  assumption 
that  it  follows  the  Mariotte-Gay-Lussac  law,  we  find 
(page  26)  1701  ccm.g. 


HEAT  AND    VOLUME  ENERGY.  85 

By  formula  (7)  we  can  also  treat  cases  of  sublima- 
tion, where  the  solid  substance  goes  directly  over  into 
the  gaseous  state.  If  we  distinguish  the  three  states 
of  aggregation  by  the  indices  I,  2,  and  3,  and  the 
transformations  between  them  by  double  indices,  then 
we  have  the  relation 


when  all  the  processes  take  place  at  the  same  temper- 
ature 6,  reversibly  and  by  constant  pressure.  From 
the  law  of  the  conservation  of  energy  it  then  follows 
that 


(10) 

For  the  change  of  volume  we  have 
(ii) 


If  we    substitute    in    (10)  the  values  from   (9),  we 
obtain 


And  from  (n)  and  (12) 
(13)  AVa 


86  PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 

Since  ^  is  the  slope  of  the  tangent  of  the  angle 

which  the  tension  curve  forms  at  the  point  considered 
with  the  B  axis,  and  can  be  considered  as  the  (nega- 
tive) slope  of  the  tension  curve,  equation  (13)  can  be 
expressed  in  words  as  follows  :  When  three  tension 
curves  meet  in  one  point  (0,  /),  the  increase  of  the 
specific  volume  by  the  increase  of  one  such  state,  is 
proportional  to  the  difference  of  slope  of  the  other  two. 

In  all  substances  the  increase  of  volume  <4F,2  by 
melting  is  small,  generally  very  small,  as  against  that 
by  evaporation  or  sublimation  ;  therefore  it  meets  the 
curves  for  the  two  latter  at  a  small  angle.  For  water, 
by  which  ^F12  is  negative,  the  sublimation  curve  at 
the  melting-point  is  steeper  than  that  for  evaporation. 

According  to  Dieterici's  results  for  water  at  o°  Cels. 
or  273°  absolute  temperature, 

^<2»  =  59^.8,  AQ,%  =  79^7  cals.  /.  J0lt  =  676.67. 

From  this  it  follows,  when  the  pressure  is  expressed 
in  millimeters  of  a  column  of  mercury, 


or 


By  which  the  specific  volume  of  steam  at  o°  Cels.  is 
204680  cm.sg.  (and  the  mechanical  equivalent  of  heat 
=  432.5).  AV^  is  smaller  than  2.io~c  times 


therefore    i~\  ,   shown   graphically,   does   not   differ 


HEAT  AND    VOLUME  ENERGY.  87 

from  oo.  Fig.  5  shows  an  approximate  picture  of 
these  relations.  The  curves  shown,  that  intersect  at 
melting-point  0  =  273°  (/  =  4.62  mm.  of  a  column  of 
mercury),  represent  the  tension  or  limiting  curves  of 


FIG.  5. 

the  three  states  of  aggregation.  In  these  curves  the 
function  V  of  d  and  /  (as  well  as  of  the  intrinsic 
energies)  is  discrete. 

If  we  lay  out,  from  the  single  points  (0,p)  of  the 
plane,  the  corresponding  values  of  V  as  perpendiculars 
to  the  plane,  we  obtain  a  surface  that  consists  of  three 
wings  which  are  independent  of  one  another;  they  cor- 
respond to  the  solid  (E),  liquid  (W),  and  gaseous  (D) 
states  of  water. 

The  well-known  phenomenon  of  the  "overcooling" 
of  water,  by  which  the  fluid  state  is  retained  below 
273°  absolute  temperature,  but  which  cannot  be 
reversibly  transformed  into  the  solid  state,  reminds 
us  that  our  results  are  only  true  when  we  consider 
reversible  changes  in  the  state  of  aggregation. 

By  equation  (i6£),  (page  78,)  we  can  conclude  at 
least,  from  the  fact  of  "  overcooling,"  that  the  chem- 
ical intensity  of  the  ice  produced  is  not  smaller  than 


88   PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

that  of  the  water  from  which  it  is  formed.  Concen- 
trated solutions  fulfil  equation  (7),  not  only  with  respect 
to  evaporation  and  freezing,  but  also  to  crystallization 
from  the  solution,  in  so  far  as  'it  is  an  isothermally  re- 
versible process. 

c.  Allotropic  Changes. 

The  transformation  from  one  allotropic  state  to 
another  takes  place,  in  many  cases,  isothermally  and 
reversibly,  just  as  with  the  case  of  a  transformation 
between  two  states  of  aggregation.  Such  changes  are 
called  by  O.  Lehmann  enantiotropic,  in  contrast  to 
monotropic,  which  take  place  only  in  one,  and  not  in 
the  reverse,  state,  as  explosive  processes. 

The  change  from  rhombic  to  monoclinic  sulphur 
takes  place  reversibly  at  95.6°  Cels.  Addition  of  heat 
increases  the  monoclinic  modification,  and  decrease  of 
heat  the  rhombic.  Here  we  have,  therefore,  a  case  in 
which  we  can  use  equation  (7) ;  it  was  proven  by  van't 
Hoff*  from  observations  by  Reicher.  From  these 
observations,  AQ  =  2.520  Cals.,  dV=-  0.014  ccm.  for 
one  gram  of  sulphur ;  it  therefore  follows  that 

Off  =  (273  +  95-6)0.014  =  ICon." 

9/          2.520.43250  g.* 

i.e.,  the  temperature  of  the  transformation  increases  by 
0.05°  Cels.,  when  the  pressure  is  increased  by  one  at- 
mosphere, which  agrees  with  our  experience. 

*  van't  Hoff,  Etudes  de  dynamique  chimique  (Amsterdam,  1884). 


HEAT  AND    VOLUME  ENERGY.  89 

d.  Dissociation. 

The  volatilization  of  ammonium  chloride  must,  al- 
though it  appears  to  be  only  a  change  of  the  state  of 
aggregation,  be  looked  upon  as  a  decomposition  of  the 
compound  NH4C1  into  NH3  and  HC1;  because  by  the 
molecular  weight,  53.5,  the  vapor  tension  should  be 
1.85,  while  at  the  temperature  of  350°  Cels.  it  is  approx- 
imately i.  The  complete  decomposition  of  the  NH4C1 
would  lead  to  the  vapor  tension  0.92,  which  lies  near  to 
the  values  observed  at  high  temperatures. 

Such  a  partial  decomposition  of  a  substance,  which 
depends  on  the  temperature,  is  known  as  dissociation 
when,  as  here,  it  takes  place  reversibly.  Since  1857, 
when  Sainte-Claire  Deville  discovered  this  phenomenon 
in  steam  and  showed  its  great  importance,  we  have 
been  able  to  bring  many  long-known  processes,  as  the 
above,  under  its  heading 

Dissociation  is  to  be  looked  upon  as  an  analogous 
process  to  evaporation  ;  for,  as  by  the  latter,  we  find  also 
by  dissociation  a  certain  pressure  by  a  certain  tempera- 
ture in  a  closed  space.  If  we  increase  the  space  in 
which  the  substance  is  confined  without  changing  the 
temperature,  the  dissociation  increases  ;  if,  however,  we 
decrease  the  volume,  the  products  of  dissociation  unite 
again,  so  that  certain  dissociation  pressure  is  always 
present  for  that  one  temperature.  For  example,  the 
dissociation  tensions  for  NH4C1  are  (Horstmann)  : 

at     200°       260°       300°       360° 

1.4         6.9        25.9       77.8  cms.of_rriercury. 


OF   THE 


9O  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

There  is,  therefore,  a  curve  for  the  dissociation 
tension,  and  the  application  of  equation  (7)  is  clearly 
apparent. 

From  the  numerous  cases  in  which  the  liberation 
of  the  water  of  crystallization  can  be  considered  as  a 
phenomenon  of  dissociation,  we  will  consider  only  the 
one  observed  by  G.  Wiedemann,  which  shows  the  de- 
pendence of  the  dissociation  tension  of  zinc  sulphate, 
ZnSO4.7H2O  upon  the  temperature  : 

at  -&  =      22°      30°      40°      50°         60°        70      78.8°       88°  Cels. 
/  =   1.97     3.15     5.49     9.29     14.88     23.3      33.78    48.67cms.  Hg. 

The  reversibility,  of  the  process  is,  however,  a  neces- 
sary condition  to  the  application  of  the  theory.  That 
calcite  at  450°  C.  gives  off  CO3  is  a  well-known  in- 
stance of  dissociation  ;  the  tension  curve  for  which  is 
given  (Le  Chatelier)  as  follows  : 

at  #  =  547°  610°  625°  740°  745°  810°  812°  865°  Cels. 
p=   27     46     56  255    289  678   753  i333mms.Hg. 

The  chemically  analogous  case  of  the  decomposi- 
tion of  manganese  carbonate  is,  however,  to  be  excluded 
here  (Ostwald),  for  the  product  manganous  oxide  and 
the  CO3  are  reduced  according  to  the  formula 

3MnO  +  COa  =  Mn,O4  +  CO, 

and  the  process  is  therefore  not  reversible. 

If  the  dissociation  consists  in  the  liberation  o/  gas- 
ecus  constituents  Irom  the  solid  or  liquid  substance, 
and  if,  further,  the  assumption  that  the  liberated  gases 
follow  the  equation  of  state  for  a  perfect  gas  is  allow- 


HEAT  AND    VOLUME  ENERGY.  $1 

able,  then  equation  (7)  becomes  essentially  trans- 
formed. As  the  increase  of  volume  we  can  then  take 
the  volume  of  the  gas  liberated,  neglecting  the  volume 
that  the  substance  occupies  when  in  the  solid  or  fluid 
state.  Since  now  the  volume  of  a  molecule  of  a  p'er- 
fect  gas  is,  at  absolute  temperature  #and  under  pressure 
/,  according  to  (5),  (page  25),  equal  to  Rfl/p\  then  the 
heat  <20,  according  to  (7),  which  must  be  supplied  for 
the  formation  of  I  mol  of  gas  is 


According  to  (8)  the  corresponding  change  of  in- 
trinsic energy  is 


,,54 


Here  v  is  the  specific  volume  of  the  gas,  and  R0  is 
equal  (nearly)  to  2  cals.  (page  27). 

Horstmann  applied  (4),  among  other  things,  to  the 
absorption  of  ammonia  by  silver  chloride.  The  com- 
pounds AgCl,  2AgCl3.NH3,  and  AgC1.3NH3  react 
towards  ammonia  as  does  a  substance  towards  its  vapor, 
in  that  by  the  pressure  of  ammonia  beside  silver  chloride, 
in  a  closed  space,  the  ammonia  is  absorbed  until  a  certain 
pressure,  depending  on  the  temperature,  is  reached. 


92  PRINCIPLES  OF  MATHEMATICAL    CHEMISTRY. 

For  example,  at  12°  Cels.  the  pressure  is  3.2  cms.  of 
mercury,  and  this  pressure  by  further  addition  of  am- 
monia remains  unchanged.  When  so  much  ammonia 
has  been  absorbed  that  the  compound  2AgC1.3NH3  is 
formed,  further  addition  of  ammonia  raises  the  pres- 
sure to  52.0  cms.  of  mercury,  and  this  remains  constant 
until  the  compound  AgC1.3NH3  is  formed.  From  the 
observations  of  the  dissociation  tension  of  these  com- 
pounds, namely, 

at          8°         12°         16°  Cels. 
/  =      4*3.2        52.0       65.3  cms.  of  Hg 

it  follows  that  for  12°  the  differential  quotient 
|S  =  2.76*.  On  the  other  hand,  it  follows  from  the 

QV 

*  Since  it  is  often  necessary  in  physical-chemical  calculations  to 
ascertain  differential  quotients  from  a  table,  the  appended  formula  is 
given.  If  we  know  the  pressure/  from  ia  to  iv  (as  from  4  to  4)  de- 
grees, that  is,,  for  0,  0  ±  w,  B  ±  2w,  etc.,  and  seek  for  0  the  differ- 
ential quotient  — ,  we  form  a  table  of  differences,  then  the  differences 
oQ 

of  these  differences,  etc.,  and  so  calculate  finally  the  arithmetical  mean 
of  the  first  and  third  of  these  differences  that  are  on  either  side  of  6. 
If  these  means  are/'(0),  f'"(B),  etc.,  then 


In  our  example  the  differences  are  8.8  and  13.3,  whose  arithmetical 
mean  is  11.05  J  if  we  now  substitute  this  in  the  formula  for/'  (6),  we 
obtain, 


|  =  -LJn.o5}=2.76. 


HEAT  AND    VOLUME   ENERGY  93 

observation  that  to  remove  one  molecule  of  NH3  the 
heat  AQ  =  9500  cals.  is  necessary,  from  (14), 

dp      9500  X  52.0 

g£  =  ^—     ~~r-  =  3.04  cms.  Hg. 

a#       2  x  285* 

That  the  two  results  do  not  agree  can  be  accounted 
for  partly  by  the  errors  of  observation,  and  partly  by 
the  use  of  (14)  instead  of  (7) ;  i.e.,  by  the  assumption 
that  the  liberated  ammonia  acts  as  a  perfect  gas. 

The  relations,  as  expressed  by  (7)  and  (14),  between 
the  tension  curve  of  a  reversible  chemical  process ;  the 
heat  necessary  for  the  completion  of  this  process ;  and 
the  changes  of  density  corresponding  to  them  ;  are  par- 
ticularly fitted  for  the  drawing  of  conclusions  from 
phenomena  which  take  place  at  very  high  or  very  low 
temperatures  and  pressures. 

We  do  not  always  need  a  supply  of  heat  for  the 
formation  of  substances  of  greater  specific  volume,  or 
finally  for  its  transformation  into  gaseous  form.  For 
only  when/  is  an  increasing  function  of  6  does  positive 
A  V  follow  from  positive  AQ.  The  dissociation  curves 
for  cyanogen,  CN-CN,  and  for  acetylene,  C2H2,  prob- 
ably, partly  rise  and  partly  fall  (Ostwald).  While  both 
substances  are  unstable  at  red  heat,  the  first  forms  in  the 
blast-furnace,  and  the  other  in  the  electric  arc.  It  is 
therefore  not  to  be  assumed  that  all  substances  on  the 
sun  are  dissociated  into  their  elements  ;  that  would  be 
true  only  for  those  whose  dissociation  tensions,  even  at 
higher  temperatures,  increase  with  the  temperature. 


94  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

e.  The  Heat  of  Solution  and  Dilution. 

Equation  (15*2),  (page  91,)  can  be  applied  to  still 
another  process  which  led  to  a  law  derived  by  Kirchhoff 
which  is  famous  as  being  the  first  application  of  the  prin- 
ciples of  thermodynamics  to  chemical  processes. 

In  order  to  evaporate  I  mol  of  pure  water  we  need 
an  amount  of  energy,  at  the  temperature  0  and  the 
vapor  tension  P0,  equal  to 


In  order  to  evaporate  the  same  amount  of  water 
from  a  solution  of  a  salt,  we  need  another  amount  of 
energy  at  same  temperature  6  and  vapor  tension 
(smaller)  P,  equal  to 

E  —  £  #2_^_//£L 


Since  in  both  cases  I  mol  of  pure  steam  is  formed 
(we  assume  here  that  the  substance  in  solution  is  not 
volatile  to  any  great  degree),  then  E  —  E0  is  the  dif- 
ference of  energy  of  I  mol  of  pure  water  and  18  rams 
of  the  solution.  If  de  is  the  amount  of  energy  which 
we  must  supply  to  18  grams  of  the  salt  solution,  to- 
gether with  the  addition  of  dN  mols  of  water  in  order 
to  keep  the  temperature  at  0,  then 


must  be  true,'  for  the  two  sides  give  the  difference  of 
energy  for  the  liquid  and  gaseous  amounts  of  water  dN. 


HEAT  AND    VOLUME  ENERGY.  95 

The  negative  heat  of  dilution  de  for  the  amount 
dN\$  therefore 


where  M  '=  inn  is  the  amount  of  water  in  grams,  and 
RQ  the  specific  gas  constant.  Since  for  highly  diluted 
solutions  de  —  o,  the  ratio  of  the  pressure  P  :  P0  is  inde- 
pendent of  the  temperature. 

This  formula  was  proven  by  Kirchhoff  *  in  1858, 
from  the  observations  of  Regnault  on  the  vapor  ten- 
sions of  different  mixtures  of  sulphuric  acid  and  water, 
and  from  those  of  J.  Thomsen  on  the  heat  of  evapora- 
tion of  the  same  mixtures,  and  the  results  were  found 
to  correspond  very  well  with  those  obtained  practically. 
The  student  is  referred  for  further  information  on  this 
subject  to  Zeuner's  Tech.  Thermodynamics,  II,  p.  38. 

*  Pogg.  Ann.  103. 


CHAPTER  VI. 

THE    RELATION     BETWEEN     HEAT    AND     ELECTRICAL. 
ENERGY. 

r* 

THE  equation  dE  =  dQ-\-dA,  which  expresses  the 
change  in  the  intrinsic  energy  E  that  takes  place  by 
the  absorption  of  heat  dQ  and  volume  energy  dA,  hold 
also,  according  to  our  principle  of  energy,  when  dA 
represents  any  other  form  of  energy.  Further,  all  the 
consequences  which  have  been  developed  in  the  pre- 
vious chapters  hold,  when  this  other  form  of  energy  is 
treated  in  a  corresponding  way  to  that  of  —  pdV,  the 
volume  energy.  The  way  of  considering  natural  proc- 
esses, which  is  known  as  energetics,  brings  into  the 
foreground  of  interest  the  similarity  of  the  different 
forms  of  energy,  and  allows  us  to  carry  over  the  con- 
clusions obtained  for  one  special  part,  to  other  parts. 

Corresponding  to  the  way  in  which  volume  energy 
was  treated,  we  can  now  treat  the  electrical  energy  of 
a  current. 

First,  we  must  recall  Ohm's  law.  If  a  current  of 
electricity  of  the  strength  of  J  amperes'*  flows  through 

*  According  to  the  international  agreement,  we  have  the  following 
definitions  for  the  practical  units: 

I  Ohm  is  equal  to  the  electrical  resistance  of  a  column  of  mercury 

96 


HEAT  AND   ELECTRICAL  ENERGY.  97 

a  conductor,  with  an  electromotive  force  equal  to  A 
volts;  and  this  conductor  passes  between  two  surfaces 
of  constant  potential  (Pl  and  P^  volts),  then 

(i)  JW=P,-P.+  A, 

where  W  is  the  constant  of  the  conductor,  lying 
between  the  two  potential  surfaces,  which  is  called  the 
resistance,  and  whose  unit  is  I  ohm.  We  will  assume 
that  the  current  goes  from  Pl  to  Pa,  i.e.,  Pl  >  Pf  If 
we  apply  this  equation  for  all  parts  of  a  simple  circuit 
and  add  them  together,  we  obtain,  since  the  final  po- 
tential of  each  part  is  equal  to  the  initial  potential  of 
the  following  one,  the  relation 

(ib)  J.^W^^A, 

in  which  "2W  is  the  total  resistance,  and  24  the  total 
E.  M.  F.  of  the  circuit. 

The  resistance  of  a  prism  of  the  length  /  cms.  and 
the  section  of  q  cms.8  is 

I  / 

w  — , 

o-  q 

where  v  is  the  specific  conductivity  and  —  the  specific 

<T 

resistance. 

106.3  cms-  l°ng  at  the  temperature  o°  Gels.,  and  whose  mass  is  14.452 
grams,  and  whose  section  is  uniformly  equal.  (. '.  i  sq.  mm.,  since 
the  sp.  gr.  of  Hg  is  13.5956.) 

I  Amplre  is  the  strength  of  a  constant  current  that  separates 
0.001118  grams  of  silver  in  i  second,  by  mean  sun  time,  from  a  water 
solution  of  silver  nitrate. 

I  Volt  is  the  potential  difference  at  the  ends  of  a  conductor  of  i 
ohm  resistance,  through  which  a  current  of  i  ampere  is  flowing,  and 
in  which  there  is  no  E.  M.  F, 


98   PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

Further,  from  Joule's  law  the  amount  of  energy  de- 
livered by  the  same  conductor  in  one  second  is 

(2)  J*  W  =  /(/>  -  P,  +  A)  volt-amperes, 

which  appears  mostly  in  the  form  of  heat ;  the  con- 
ductor which  takes  the  current /from  potential  P2  to 
P,  again  would,  were  no  E.  M.  F.  present,  give  up,  in 
the  second,  the  amount  of  energy 

(2b)  /(/>,  -  />,)  volt-amperes. 

In  the  whole  circuit,  therefore,  the  E.  M.  F.  A  causes 
the  delivery  of  energy  to  the  amount 

(2c)  J .  A  volt-amperes 

in  i  second,  or  in  time  dt  the  amount 

(3)  J .  A  .  dt  volt  X  amperes  X  seconds. 

Since  I  volt  =  io8  absolute  units  of  electrical  poten- 
tial, i  ampere  =  i/io  of  an  absolute  unit  of  current 
strength  ;  then  (page  21) 

i  volt  X  ampere  X  second  =  io7  erg  =  — —  Cals. 

4.24 

—  0.24  cals.; 

I  volt  X  ampere  =  i  watt  =  0.24  — -", 

sec. 

Finally,  if  we  designate  the  product  Jdt  as  the 
amount  of  electricity  in  the  conductor  during  time  dt, 
so  that  according  to  the  usual  way  of  consideration, 
one  half  of  this  amount,  as  positive  electricity,  goes 


HEAT  AND   ELECTRICAL   ENERGY.  99 

through  the  conductor  in  one  direction,  and  the  other 
half,  as  negative  electricity,  goes  in  the  opposite  one, 
and  place 

(4)  Jdt  —  de, 

we  obtain  for  the  equation  of  energy,  for  the  portion 
of  the  conductor  considered, 


(5) 


Here  E  is  the  intrinsic  energy,  0  the  absolute  tem- 
perature, and  »£  the  entropy  of  that  portion  of  the  con- 
ductor that  lies  between  the  potential  surfaces.  Ade  is, 
as  already  remarked,  the  total  energy  which  comes  from 
the  E.  M.  F.  for  that  portion  of  the  conductor,  and 
can  therefore  also  appear  partly  as  Joule's  heat,  devel- 
oped in  the  conductor.  Finally,  6dS  expresses  the 
change  of  heat  from  other  sources  in  the  conductor. 

Equation  (5)  stands  now,  in  its  form,  and  in  the 
physical  meaning  of  its  quantities,  in  such  complete 
analogy  to  (3)  on  page  80  that  all  the  conclusions  ob- 
tained there  can  be  carried  over  to  it.  Of  xourse  A 
here  is  not,  as  p  there,  an  intensity  quantity  (page  43), 
but  is  the  difference  between  two  such  quantities,  i.e., 
the  potentials.  If  now,  in  general,  we  designate  the 
potential  by  P,  the  second  member  of  the  equation 
falls  into  two  members  of  the  form  Pde  that  stand  in 
energetical  analogy  to  —  pdV  (IdM  on  page  43);  In 
fact,  an  electrical  current  is  only  possible  between  two 
places  of  different  potential,  and  the  total  amount  of 
electricity  is  an  unvarying  quantity.  Here  we  can 


IOO  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

liken,  as  is  usual,  the  electrical  energy  to  a  water-pipe 
connected  with  a  pump,  i.e.,  represent  electrical  energy 
as  volume  energy. 

The  method  of  reaching  conclusions  that  was  used 
on  page  80  leads  now  from  equation  (5)  to  the  law 

(6)  4Q=V.~.Ae. 

QV 

If  we  change  the  E.  M.  F.  reversibly  with  the  tem- 
perature, then  for  the  discharge  of  electricity  equal  to 
At  through  a  conductor,  in  which  the  E.  M.  F.  as  well 
as  the  temperature  6  is  kept  unchanged,  we  need  a 
supply  of  heat  equal  to  AQ,  which  is  proportional  to  6 
and  to  Ae,  besides  being  proportional  to  the  fall  of 
potential. 

According  as  A  is  an  increasing  or  a  decreasing 
function  of  8,  heat  must  be  supplied  or  withdrawn,  in 
order  that  the  E.  M.  F.  does  work,  without  change  of  tem- 
perature ;  that  is, 


This  law  (6),  discovered  by  v.  Helmholtz,  holds  for 
all  reversible  transformations  of  heat  into  current  en- 
ergy, for  example,  also  for  thermo-elements  ;  but  here 
we  will  follow  out  only  the  case  for  galvanic  batteries. 

In  these  the  energy  caused  by  the  E.  M.  F.  is 
occasioned  by  chemical  changes  that  take  place  in  the 
cell,  According  to  Faraday's  electrolytic  law,  there 
wanders  from  the  anode  to  the  kathode,  with  the 


HEAT  AND   ELECTRICAL  ENERGY.  IOI 

amount  of  electricity  I  ampere  X  second  =  i  coulomb, 
a  determined  amount  of  hydrogen,  namely, 

—  =  #o  =  0.00001036  grams, 

^0 

whenever  H  is  electrolytically  separated.  In  general, 
there  will  be  separated  of  the  chemical  elements  present 
an  amount  equal  to 

—  =  xjx.  =  0.00001036.  a  grains. 

^0 

where  a  is  the  chemical  equivalent  weight  of  the  ele- 
ment considered,  x^  is  found  to  be,  from  the  electro- 
lytic separation  of  silver,  0.001118:107.938.  The 
equivalent  weights  are  found  from  the  atomic  or 
molecular  weights  by  division  by  small  whole  numbers, 
the  valences ;  for  example, 

for      H     Cl          O  N 

equivalent  =  i     35.5     8  =  -      4.67  =  -^. 

Some  elements,  as  mercury,  have  different  equiva- 
lents in  different  compounds  ;  for  example,  for  Hg  in 

HgCla  we  have  -22l  t  while  in  HgCl  we  have  199.8. 
Since,  however,  by  the  same  current, 


102   PRINCIPLES  OF  MATHEMATICAL    CHEMISTRY. 

are  decomposed,  we  can  say  that  for  equal  currents,  in 
the  same  time,  the  same  number  of  valences  are  dis- 
solved. 

j 

With    I   equivalent  there  wanders   always  e0  =- 

•*« 

=  96,540  coulombs. 

When  the  chemical  process  which  takes  place  in  the 
cell  is  so  carried  out  that  it  can  cause  no  current,  as  in 
a  calorimetric  bomb,  then  the  change  of  intrinsic 
energy,  which  takes  place  in  the  bomb,  is  shown  wholly 
as  heat.  We  will  assume  that  by  the  decomposition  of 
the  number  of  grams  of  the  electrolyte,  as  is  expressed 
by  the  equivalent  weight,  the  decrease  of  the  intrinsic 
energy  AE  is  measured  by  q  cals.  Then  by  the  dis- 
charge of  I  coulomb  the  intrinsic  energy  of  the  cell 
sinks  by  xq  cals.,  and  equation  (7)  gives 

—  xq  cals.  =  f^-g  —  A J  volt-amperes-seconds 

—  4.24  X  0.00001036?  =  6— -  —  A  cals ; 

30 


0.00004393? 
(8>    a=  22800  (A  -  *|J)  cals.  =  23(A  -  6  £=]  Cals. 


The  older  theory  of  the  galvanic  element  left  out 
this  second  member  of  the  right-hand  side  ;  as  observed, 
this  was  rectified  by  v.  Helmholtz.*  The  heat  of  reac- 

*v.  Helmholtz,  Zur  Thermodynamik  chemischer  VorgSinge  1882, 
1883,  in  Ges.  Abhandl.  Gibbs  recognized  before  that  (1878)  the  con- 
nection between  heat  and  current  energy. 


HEA  T  AND   ELECTRICAL  ENERG  Y.  103 

tion,  produced  by  the  decomposition  of  a  chemical 
equivalent  weight,  is  therefore  greater  or  smaller  than 
the  E.  M.  F.,  expressed  as  heat,  according  as  this  is  an 
increasing  or  a  decreasing  function  of  the  temperature. 

For  the  proof  and  use  of  the  theory  the  following 
examples  will  suffice.  They  are  taken  principally  from 
the  observations  of  Jahn.* 

The  E.  M.  F.  of  a  Daniell  cell  is  A  =  1.0962  volts, 
corresponding  to  1.0962  X  23  =  25.2  Cals.  On  the 
other  hand,  it  is  observed  that,  by  the  transformation 
of  i  mol  or  2  equivalents,  the  intrinsic  energy  of  the  cell 
is  decreased  by  50.1  Cals.,  therefore  by  transformation 
of  I  equivalent  by  25.05  Cals.  By  the  formation  of  a 
half  mol  of  ZnSO4  from  Zn,  O,  and  highly  diluted 
HaSO4  there  are  53045  Cals.  developed,  and  by  the  for- 
mation of  a  half  mol  of  CuSO4 ,  correspondingly, 
27980  Cals.  The  difference  is  25.065  Cals.  The  small 
difference  between  this  calorimetrical  and  the  electrical 
determination  leads  us  to  suspect  that  the  E.  M.  F.  is 
almost  entirely  independent  of  the  temperature.  In 

QJ 

fact  Gockel  observed  •=-*  =  0.000034  volts,  which  cor- 
responds to  23  X  273  X  0.000034  =  0.21  Cals. 

A  greater  influence  of  the  temperature  coefficient  is 
shown  in  the  following  cases  : 

An  element  consisting  of  lead  and  copper,  which 
are  suspended  in  their  acetates,  and  separated  from 
each  other  by  a  clay  cell,  gave  the  E.  M.  F.  0.47643 
volts  =  to  10.96  Cals.  for  I  equivalent,  or  21.9  Cals.  for 

*Wied.  Ann.  28- 


IO4  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY, 

I  mol.     Since  we    observe    calorimetrically  only  16.5 
Cals.,  the  remaining  5.4  Cals.  must  be  taken  from  the 

surrounding    objects.       From    the    observation,  —„  = 

0.000385  volts,  it  follows  23  X  273  X  0.000385.2  =  4.8 
Cals.,  which  is  taken  up  as  above  mentioned. 

If,  on  the  other  hand,  silver  and  lead  are  present  in 
their  nitrates,  the  E.  M.  F.  0.932  volts  leads  to  42.9 
Cals.,  while,  calorimetrically  measured,  we  find  50.9 
for  the  transformation  of  I  mol.  Here  q  is  greater 
than  23 A9  i.e.,  during  an  isothermal  action  the  cell  will 
give  off  heat  to  surrounding  objects,  or  when  this  is  not 
possible  to  give  it  off,  it  will  be  absorbed  by  the  cell, 
and  this  is  consequently  heated,  The  absorption  of 

heat   amounts   here   to  —  8   Cals.,   while   from   ^.= 

OV 

0.00063235,  —  7.9  follows 

From  the  results  of  the  "  physikalisch-technischen 
Reichsanstalt,"  the  E.  M.  F.  of  a  Clark*  cell,  between 
10°  and  30°  Cels.,  is 

A  =  1.438  —  o.ooi  i6($  —  1 5)  —  o.ooooi($  —  1 5)' 

How  many  per  cent  of  its  E.  M.  F.  causes  the  ab- 
sorption of  heat?  From  (6)  it  follows  that  the  heat 
absorbed  by  passage  of  I  coulomb  is 


*The  Clark  cell  consists  of  Hg  and  Zn  separated  by  a  paste  of 
HgSO4 ,  the  Zn  being  in  a  saturated  solution  of  zinc  sulphate,  and 
the  whole  made  air-tight  with  paraffine. 


HEAT  ANL>   ELECTRICAL  ENERGY. 
therefore  its  ratio  to  the  E.  M.  F.  at  15    Cels.  is 

AQ      VdA       288 

~  aooil6>  =  -  °'23  ; 


i.e.,  23^  of  the  E.  M.  F.  is  given  up  to  surrounding 
objects.  The  chemical  process  gives  123$. 

An  important  law,  concerning  the  process  in  a  gal- 
vanic battery,  can  still  be  derived,  from  this  manner  of 
development,  when  we  apply  the  principle  of  the  con- 
servation to  a  portion  of  the  electrolyte  lying  between 
two  potential  surfaces.  In  this  case,  besides  the  changes 
of  heat  and  current  energy,  considered  in  (5),  we  must 
also  bring  in  the  changes  of  substances,  as  possible 
changes,  as  was  done  on  page  75. 

We  will  restrict  ourselves  to  the  case  of  reversible 
changes  and  stationary  state  of  the  current.  Let  on 
the  entrance-surface  of  the  current  the  amount  of 
matter  dM,  whose  chemical  intensity  is  IT,  (unvariable), 
be  taken  up  ;  on  the  other  side  let  the  same  dM,  with 
the  chemical  intensity  -7Ta  ,  be  carried  farther.  If,  further, 
between  the  two  potential  surfaces  the  E.  M.  F.  A  is 
active,  then  the  amount  of  energy  used  up  by  the 
transmission  of  the  amount  of  electricity  de  is  Ade. 
Finally,  it  follows,  from  the  condition  that  the  state  is 
now  stationary,  that  no  change  of  entropy  or  energy 
takes  place,  so  that 

(9)  dE  =  o  =  -  Ade  +  HJM  —  HJM. 

Electrical  exchanges  have,  however,  the  property 
in  electrolytes,  that  with  a  certain  amount  of  elec- 


106  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

tricity  goes  a  certain  amount  of  matter  ;  namely,  ac- 
cording to  page  70, 


(10) 

Therefore  it  follows  that 
(gb)  <2E  =  o  =  [(  IT,  —  JT,  )x,a  -  A~]de, 

an  equation  that  can  hold  only  when  the  transmission 
is  reversible  and  stationary,  when 


The  difference  of  the  electrical  potential,  caused  by  the 
reversible  stationary  chemical  process  which  is  present  in 
the  two  surfaces  of  constant  potential,  is  proportional  to 
the  difference  of  chemical  intensities,  measured  in  equiva- 
lents of  these  'surfaces.  Since  now  every  chemical 
reaction  is  occasioned  by  intensity  differences,  then,  as 
will  be  shown  later,  every  electrometer  is  a  chemometer. 
(Ostwald.) 

At  any  rate,  by  the  foregoing  considerations  we  can 
perceive  the  true  nature  of  the  E.  M.  F.  from  chemical 
sources  ;  it  appears  to  be  equivalent  to  the  loss  of 
chemical  intensity,  as  the  energy  of  the  electrical  cur- 
rent is  equivalent  to  the  chemical  energy. 

According  to  equation  (n),  with  the  aid  of  (i),  we 
can  place 

(I  ib)  xjla  =  Pe0  =  Ua  ; 

i.e.,  II  suffers  a  strain  when  P  does,  and  therefore  so 
does  every  seat  of  the  E.  M.  F.  If  by  dE  we  understand 
the  amount  of  energy  which  goes  through  a  certain 


HEAT  AND   ELECTRICAL   ENERGY.  IO? 

section,  and  under  dS,  dM,  and  de  the  amounts  of  en- 
tropy, matter,  and  electricity  which  wander  with  it,  then 

(12)  dE  =  VdS  -  Pde  -f-  UdM. 

It  remains  now  only  to  bring  the  conception  of 
conductivity  more  exactly  into  view,  and  to  join  it  to 
the  ideas  of  the  transportation  of  electricity  and  of 
matter,  which  takes  place  during  electrolysis. 

The  strength  of  the  current  that  flows  in  direction 
x  through  a  prism  of  the  length  dx,  with  the  specific 
conductivity  <r  and  the  section  <?,  is  equal  to,  according 
to  Ohm's  law,  when  for  the  length  dx  the  potential 
increases  by  dP, 

dP 

(13)  '=:r:*5* 

The  amount  of  electricity  which  goes  through  the 
section  in  a  second  is 

dP 
(13*)  ^  =  -Tx 

With  this  amount  of  electricity  x^aj,  grams  of  the 
anion  goes  to  the  kathode,  and  xjxj  grams  of  the 
kation  goes  to  the  anode,  and  x^ai  grams  of  the  elec- 
trolyte is  decomposed  in  the  second. 

If  the  anion  wanders  with  the  velocity  u,  and  the 
kation  with  the  velocity  u^ ,  and  if  .Afy,  of  the  N  equiva- 
lents present  in  a  cubic  centimeter,  are  electrolysed,  or 
separated  in  their  ions  (i.e.,  if  there  are  Nrf  equivalents 
of  each  ion  in  I  cc.),  then  Nrful  anion  equivalents  and 
kation  equivalents  wander  in  the  second,  on 


108   PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 


which  there  is  bound  Nijujx^  and  Nyu^/x^  coulombs 
of  electricity  respectively. 

If  now  we  accept  the  hypothesis  of  F.  Kohlrausch* 
that  these  currents  are  independent  of  one  another 
and  each  moves  under  the  influence  of  the  above  po- 
tential difference,  then 


where  the  hypothetical  specific  conductivities  <r4  and  <ra 
are  to  be  so  chosen  that 


while 


is  the  total  amount  of  electricity  in  motion.  If  we 
call  the  conductivity  of  each  equivalent  present  //, 
so  that 


where  now  p  is  the  molecular  conductivity,  and  trans- 
form the  hypothetical  conductivities  of  the  ions  in  a 
corresponding  way, 

Wjf  =  o-,  ,     r]^N  =  <rf  ,     rj(^  -f-  ^  =  //, 
*Wied.  Ann.  6,  1876. 


then 


(18) 


HEAT  AND   ELECTRICAL   ENERGY.  109 


77 


(19) 


The  hypothetical  molecular  conductivities  of  the 
ions  are  therefore  proportional  to  their  wandering  ve- 
locities. From  (17)  and  (19)  it  follows  that 


(20) 


therefore  by  total  dissociation  into  ions,  i.e.,  for  77  =  I, 
where  /^  is  the  value  of  the  molecular  conduc- 
tivity, we  have 


F.  Kohlrausch  gives  as  molecular  conductivity  ju*; 
the  specific  conductivity  of  a  substance,  with  mercury 
as  a  standard,  divided  by  the  number  of  gram  equiv. 
alents  in  a  liter.  That  is, 


JW*  =  :  I0007V, 


110  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

where  0"Hg  is  the  specific  conductivity  of  mercury.    By 
the  aid  of  (17)  we  then  have   . 

1000  „.  looo 

" 


The  specific  conductivity  may  be  found  from  the 
definition  of  the  ohm.  It  is,  in  absolute  units, 
<rHg  =  1.063  .  io-5. 

The  ratio  of  the  wandering  velocities  can  now  be 
obtained,  according  to  Hittorf,  from  the  changes  in 
concentration  of  the  electrolyte  at  the  electrodes. 
The  wandering  of  Nrju^  equivalents  of  the  anion 
means  that  at  the  kathode,  in  the  second,  each  cubic 
centimeter  loses  the  amount  of  NrfU^  equivalents  of 
the  electrolyte,  these  equivalents  being  separated  out. 
If  a  is  the  equivalent  weight  of  the  electrolyte,  al  that 
of  the  anion,  and  a^  that  of  the  kation,  then  the  fol- 
lowing scheme  gives  the  number  of  grams  in  the 
second  that  wander  or  are  separated  : 

Anode.  Kathode. 


Separation. 


The  total  amount  separated  is  Nrfa(u.t  +  wa),  which 
stands  in  the  following  proportions  to  the  decrease  of 
concentration  which  accompanies  it  : 


Z  -  *  *         7  - 

3  l 


HEAT  AND    ELECTRICAL   ENERGY.  Ill 

For  example,  Hittorf  passed  a  current  of  0.0626 
amperes,  for  four  hours,  through  a  copper-sulphate 
solution  which  contained  2.8543  grams  of  CuO,  as 
determined  for  a  certain  volume  by  KOH.  After  the 
current  had  passed  through,  the  same  volume  of  solu- 
tion, taken  from  the  neighborhood  of  the  kathode, 
gave  only  2.5897  grams  of  CuO,  and  the  loss  of  copper, 
corresponding  to  0.2646  grams  CuO,  is  O.2II2  gram. 
This  number,  by  a  simple  calculation,  will  give  the 
value  of  */,<*„.  During  the  passage  of  the  current 
0.2955  gram  is  separated  as  kation,  from  which,  by 
the  same  calculation,  we  can  find  (ut  -f-  #a)ov  From 
the  anode  the  amount  0.2955  —  0.2112  =0.0843  gram 
Cu  is  carried  over.  The  quotient, 


=a2g 

*!    +    «,  0.2955 

is  called  by  Hittorf  the  transference  number  (Ueber- 
fuhrungszahl)  of  the  kation.  The  value  following 
from  this, 

*,  =    -         =1-0.285  =0.715, 


is  the  transference  number  of  the  anion,  and  gives  the 
amount  of  the  anion  ula1  which  has  wandered,  since 
during  the  time  of  observation,  in  each  second,  «x-f-  U2 
equivalents,  or  (#,  +  #a)<*i  grams,  of  the  anion  SO4 
must  be  separated. 

In  another  experiment  a  current  of  0.1428  ampere 
was  employed  for  two  hours,  and  0.3372  gram  of  Cu 


112   PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

were  separated  on  the  kathode.  The  solution  around 
the  kathode  gave  before  the  experiment  2.8543  grams 
and  afterward  2.5541  grams  of  CuO,  so  that  there  was 
a  loss  of  0.3002  gram  of  CuO,  corresponding  to  0.2396 
gram  of  Cu.  The  amount  transferred  is  there- 
fore 0.0976  grams  Cu,  and  the  transference  number 

of  the  kation  is  -  ~ — -  =  0.289,  and  that  of  the  anion 
0.3372 

0.711 — which  corresponds  with  the  other  experiment. 

For  highly  diluted  solutions  Hittorf  found  the 
transference  number  for  Na  =  0.380,  and  Kohlrausch 
found  the  limit  (grenzwerth)  of  the  conductivity  (18°) 

/**=  102.8  X  io-7. 
From  the  relations 

ir* 

•    g.  J- — ^  =  0.380,     Mi    +  /V  =  102.8, 

the  hypothetical  conductivities  are  found  to  be 
p*  =  63.7 .  io~7,     /*„  =  39.1 .  io~7, 

which  are  proportional  (19)  to  the  wandering  velocities. 
For  KC1  in  the  same  way 


*=  121.6.  iar'; 


therefore, 

/i,  =  61.9  .  io-7,     p*  =  39.7  .  io-7. 

The  chlorine  has  in  both  cases  approximately  the 
same  value,  an  average  of  62.8,  which  is  also  supported 


HEAT  AND   ELECTRICAL  ENERGY.  113 

by  other  electrolytes.  The  still  present  uncertainties 
in  the  conductivities  are  decreasing  by  continued  care- 
ful work. 

The  conductivity  of  silver  has  been  very  carefully 
determined  by  Loeb  and  Nernst  for  numerous  cases 
and  found  in  average  to  be  59.1.  AgNO3  gave  59.2, 
AgClO3  58.5  at  25°  Cels.  The  latest  collections  of  the 
conductivities  can  be  found  in  Wied.  Ann.  50.  (Kohl- 
rausch.)* 

*  See  also  Ostwald's  Lehrbuch  der  allg.  Chem.  II.— TRANS. 


PART   III. 
THE  CHEMICAL  INTENSITY. 


CHAPTER   I. 

THE  GENERAL  PROPERTIES  OF  CHEMICAL   INTENSITY. 

IT  is  now  time  to  show  how  we  are  to  treat  changes 
of  energy  that  are  caused  by  an  exchange  of  substance. 

If  the  energy  of  a  body  changes,  not  only  by 
absorption  or  loss  of  heat  and  volume  energy,  but  also 
by  gain  or  loss  of  mass,  then  the  differential  of  the 
energy  is  given  by  the  formula 


(i)    dE 

...     nndM. 


Here  E,  S,  and  V  are  respectively  the  intrinsic 
energy,  entropy,  and  volume  of  the  body,  9  its  tem- 
perature (absolute),  and/  the  pressure  under  which  it 
is.  Further,  M19  M^  .  .  .  Mn  are  the  masses  of  the 
chemical  constituents  of  the  bodies  which  are  in  the 
state  of  exchange,  and  77,,  JI3,  .  .  .  Tln  their  chemical 

intensities, 

114 


PROPERTIES   OF  CHEMICAL   INTENSITY.       11$ 

That  the  function  TI — which  expresses  mathemati- 
cally the  differential  quotient  of  the  function  E  with 
respect  to  M—  possesses  the  physical  properties  to 
entitle  it  to  the  name  of  an  'intensity  quantity,  was 
shown  for  a  simple  case  on  page  78.  The  general 
proof  follows  below.  It  is  not  possible  now,  as  then, 
to  consider  only  two  phases,  for,  in  general,  it  is  not 
possible  to  realize  this  without  having  at  the  same  time 
exchanges  with  other  phases. 

We  imagine  a  system  of  m  phases,  i.e.,  bodies  that 
are  in  a  state  of  exchange,  and  let  each  phase  contain 
all  or  some  of  the  n  substances  whose  total  masses,  in 
the  whole  system,  are  given  by  Mlt  M9t  .  .  .  Mn. 

The  changes  of  these  substances  we  will  assume 
to  be  independent  of  each  other,  i.e.,  an  increase  or 
decrease  of  one  is  possible  without  a  change  in  the 
amounts  of  the  others.  Each  phase  is  therefore  a 
mixture  and  not  a  chemical  compound  of  these  n  sub- 
stances. If  we  distinguish  now  the  values  of  the  func- 
tions which  belong  to  each  phase  in  equation  (i)  by 
accents  (as  ',  " ,  '" ,  etc.),  then  the  energy  equations  for 
the  single  phases  are 


dE'<erdS—p'dV'  +  I. 
-f-7 
dE"  5  Q"dS"-p"dV"-\-  U^'dM" 


And  there  are  as  many  equations  as  there  are  phases, 
m,  present.  The  m  phases  are,  however,  to  be  so 
chosen  that  no  exchange  of  matter  takes  place  except 


Il6  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

between  them,  and  that  changes  of  heat  and  volume 
take  place  only  inside  the  system,  as  it  does  in  case  of 
equilibrium.  The  system  is  therefore  isolated,  and  no 
changes  of  energy  are  possible  but  the  internal  ones 
which  occur  between  the  phases.  Then 

(3)  dE  =  dE'  +  dE"  +  .  .  .  =  o. 

" 

(  dM,  =  dM;  +  dM9"  +  .  .  .  =  o, 
(  dM,  =  dMj  +  dM9"  +  .  .  .  =  o. 

These  equations  are  to  be  considered  as  limits  to  the 
otherwise  arbitrary  differentials  which  occur  in  (2)  ;  and 
we  will  now  treat  the  case  further  under  the  condition 
that  no  other  limits  exist  but  these,  which  follow  from 
the  conception  of  an  isolated  system. 

If  we  add  the  equations  in  (2),  it  follows  from  (3) 
that 

f  o  <  8'dS'+o"dS"+  .  .  . 
-p*dV'-p"dV"-..  . 


Now  we  can  express  each  of  the  quantities  in  equa- 
tions (4)  and  (5)  by  the  m  —  I  others  ;  for  example, 

W  =  -dV"  -dV"  -  ... 
Accordingly  we  can  re-form  (6)  into 


fThis  sum  does  not,  in  all  cases,  equal  the  entropy  of  the  system 
(see  page  70). 


fV        OFTHK      "TKV 

I  UNIVERSITY  } 


PROPERTIES  OF  CffEM^^^S^^gfT  Y.       II  ? 

When,  in  this  manner,  we  transform  all  the  con- 
stituents of  equation  (6),  then  the  differentials  are 
completely  independent  of  one  another,  provided  that 
there  are  no  other  conditions  than  those  expressed  by 
equations  (4)  and  (5).  We  can  therefore,  for  example, 
place  all  the  differentials  up  to  dV"  equal  to  zero  ;  we 
then  find,  since  dV"  can  assume  any  positive  or  nega- 
tive infinitely  small  value,  that  it  is  necessary  for  the 
truth  of  equation  (6)  that  p"  —  /'  =  o.  From  this 
well-known  method  of  conclusion  it  follows  that 

(  0'  =  0"  =  8'"  =  ...  =  8, 
}/=/'=/"  =...=/. 


77'  =  77"  =  77'"=.,  .  =  77. 


2  — a  — a- 


When  the  temperatures  and  pressures  of  all  the 
phases  of  an  isolated  system  are  equal,  then  the  chemical 
intensities  of  all  the  constituents  are  equal  also,  when 
there  is  no  other  condition  to  the,  exchange  of  matter  than 
the  law  of  tJie  conservation  of  matter. 

This  law  has  only  one  exception,  i.e.,  in  cases  where 
it  is  not  possible  for  a  positive  and  a  negative  change 
of  the  differential  to  take  place.  The  volume  and  en- 
tropy of  a  phase,  as  well  as  the  masses  of  its  single 
constituents,  can,  under  all  circumstances,  increase  or 
decrease,  and  we  will  assume  here  that  new  phases 
appear  in  the  process ;  or  that  during  it  new  constit- 
uents enter  into  them  which  were  previously  not  there. 
The  first  case,  that  of  the  formation  of  a  new  phase, 
cannot  be  reconciled  to  the  conditions  that  are  ex- 
pressed for  the  differentials  by  (3),  (4),  and  (5),  and 


Il8   PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

must  therefore  be  excluded.  On  the  other  hand,  new 
constituents  can  enter  into  a  phase  during  the  process, 
as,  for  example,  in  phase  Ml  ,  by  which  its  differential, 
in  the  example  dM'>  can  assume  only  positive  values. 
Since  by  this  the  other  differentials  in  equation  (6)  can 
be,  placed  equal  to  zero  up  to  dM"  (or  dM^",  etc.), 
which,  according  to  (5),  must  be  chosen  as  negative, 
we  have 


i.e.,  the  intensities  of  the  constituents  in  the  phases 
which  are  newly  added  can  not  be  smaller  than  the 
others.  In  the  example 

(;*)  H^U",    77,'  >  77,'",  etc. 

It  is  necessary  to  add  to  this  proof,  originated  by 
W.  Gibbs,f  still  the  following  : 

Equation  (2)  gives,  with  help  of  (3),  (4),  and  (5), 

dE^o. 

And  in  the  case  that  relation  (jb)  is  true,  this  inequality 
is  the  more  fulfilled. 

It  agrees  with  (i),  in  case  dS  =  dS'-\-  .  .  .  ,  only  when 
the  equality  sign  is  true  in  (i),  i.e.,  when  the  process  is 
reversible.  Only  reversible  processes  allow,  therefore, 
the  assumption  of  the  conservation  of  entropy  In  an 
isolated  system  ;  if  in  such  a  system  non-reversible 
processes  take  place,  then  it  is  not  possible  to  exclude 

f  W.  Gibbs,  Thermodynamische  Studien,  German  by  Ostwald, 
Leipzig,  1892  ;  or  Trans,  of  the  Conn.  Academy,  vols.  n  and  in, 
1873-78. 


PROPERTIES   OF  CHEMICAL   INTENSITY.       119 

the  case  in  which  the  entropy  of  the  system  varies 
from  the  sum  of  the  entropies  of  the  parts. 

The  functions  can  be  found  in  the  case  of  a  gas 
mixture,  since  for  this  the  energy,  entropy,  and  volume 
are  known  as  functions  of  the  pressure  and  tempera- 
ture. If  we  apply  equation  (i)  to  a  gas  mixture,  we 
understand  by  dE  the  change  that  the  intrinsic  energy 
suffers  when  changes  of  heat,  volume,  and  amounts, 
of  the  single  gaseous  constituents,  are  possible.  We 
write  (i)  in  the  form 

(8)  dE  -  6dS  +  PdV^  UldMl  +  HJM^  ...+  HndMn, 

in  which  we  represent  by  P  the  total  pressure  (page  68) 
of  the  gas  mixture  to  distinguish  it  from  the  partial 
pressures/!,  /a,  .  .  .  pn  of  the  single  constituents. 

By  equation  (13)  (page  52)  the  energy  for  the  unit 
of  weight  of  a  gas,  whose  specific  heat  by  constant 
volume  is  cv,  at  the  absolute  temperature  8,  equals 

(9)  t  =  e.  +  c,8 

when  e0  is  a  constant.  The  amount  M  grams  of  this 
gas  possesses,  then,  the  energy 

Me  =  Me,  +  Mcv0y 
whose  differential  is 


If  we  apply  this  to  all  the  single  different  constitu- 
ents of  the  mixture,  distinguished  by  indices,  the 
energy  differential  of  the  mixture  follows,  as 

(10)  dE  =  (e,dM,  +  e,dM9  +  .  .  .) 


120  PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 

On  the  other  hand,  equation  (14)  (page  70)  gives 
the  entropy  of  the  unit  of  weight  of  a  gas  whose  vol- 
ume is  F,  and  whose  gas  constant  is  R,  as 

(i  i  )  s  =  s0  +  RIV+  cJ6R  -  RIM. 

The  amount  of  M  grams  of  this  gas  has  then  the 
entropy  Ms,  the  differential  of  which  is 


+  Mc-     ~  RdM+sdM. 

Under  the  condition  that  the  mixture  of  the  single 
gases  is  a  reversible  process,  the  addition  of  the  single 
entropies  gives  the  total  entropy.  The  total  entropy 
of  the  whole  system  changes,  therefore,  by  the  amount 


If  we  transform  this  equation,  remembering  equa- 
tion (7)  (page  68),  we  obtain 

(13)  PF=(M1R1  +  M^  +  ..  .)B, 

and  by  substitution  of  these  values  (10)  and  (12)  in  (8) 
we  find,  since  the  expressions  multiplied  into  ^/Fand 
dO  cancel,  and  the  left  side  of  (8)  contains  only  the 
differentials  dM  (as  it  must  in  consequence  of  the  right 
side), 

(14)  (*,  -  6st  +  Rte  dM  +  (*,-  6s.  + 

^  Tl.dM,  +  UJM%  +  .  .  . 


PROPERTIES  OF  CHEMICAL   INTENSITY.       121 

If,  now,  the  terms  dM  are  entirely  independent  of 
one  another,  i.e.,  an  actual  gas  mixture  is  considered, 
whose  each  constituent  can  increase  or  decrease  en- 
tirely independent  of  the  others,  it  follows  from  (14) 
that 


(15) 


In  consequence  of  the  equation  of  state  of  a  gas 
when  the  specific  volumes  are  vlt  vtt  .  .  .  ,  we  have 


i.e.,  the  chemical  intensities  of  each  independent  con- 
stituent of  a  gas  mixture  is,  in  reversible  changes,  equal 
to  the  thermodynamical  potential  of  the  unit  of  weight 
of  this  constituent,  by  constant  pressure  ;  otherwise  it 
is  greater.  From  the  consideration  of  equations  (9) 
and  (u),  and  the  relation  between  the  gas  constant 
and  the  specific  heats, 

(16)  \p  -  c,  =  R, 
we  find  further  that 

(17)  n^e^  +  cPld-S^0-6R,lV 


and  the  corresponding  values  for  7I2  and  7T8  .  .  .  ,  which 
we  will  express  in  general  (i.e.,  without  indices),  are 
given  by 


>  e0  +  cpB  -  soie  -  6RIV-  cJBlQR  +  BRIM. 


122    PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

If  instead  of  equation  (14),  page  70  (above  equation 
(u)),  we  use  (13),  on  same  page,  for  the  transformation 
of  (15),  we  obtain 

(i  8)    n^e.  +  ^o-  sQfre  -  cpeie  +  RSIP  +  RBIC, 
+  cje  —  s0e  -  CPWR  +  RBIP  +  ROIC, 


where  C  is  the  concentration  of  the  constituent  to 
which  the  function  71  belongs.  Since  C  —  p/P  (equation 
(9),  page  69),  it  follows  that 


(19)       ne0  +  cfe  —  s0e  — 

By  differentiation  of  (18)  we  find 


i.e.,  with  increasing  concentration  of  a  gas  in  a  mix- 
ture, when  the  temperature  does  not  change,  the 
intensity  of  the  gas  increases  the  slower,  the  greater 
the  already  present  concentration  is. 

The  law,  proven  on  page  117,  of  the  equality  of  the 
chemical  intensities  of  the  different  phases  of  the  same 
substance,  allows  us  to  draw  into  consideration  the 
chemical  intensities  of  other  substances  that  are  not 
gaseous.  We  will  assume  any  solid  or  liquid  body 
that  contains  one  constituent  that  is  independent, 
which  is  present  in  the  concentration  C.  If,  now,  there 
is  a  gas  mixture  which  contains  the  same  independent 


PROPERTIES  OF  CHEMICAL   INTENSITY.        123 

constituent  in  the  same  concentration  C,  at  the  same 
temperature  and  pressure,  that  is  in  contact  with  the 
solid  or  liquid,  so  that  they  form  together  an  isolated 
system,  then  the  constituent  considered  has  the  same 
intensity  in  the  gas  mixture  and  in  the  body.  This 
intensity  is 

(21)  n>II*  +  R6lC, 

where  R  is  the  specific  gas  constant  for  the  constituent, 
and  II*  that  function  of  6  and  P  which  the  intensity  of 
the  constituent  would  give  were  it  alone,  i.e.,  in  con- 
cenntration  I. 
According  to 


(22)  II*  =  e*  -  6s*  -f  Pv* 

where  e*,  s*  and  v*  are  the  values  of  the  energy, 
entropy,  and  volume  of  one  gram  of  the  gaseous  con- 
stituent, which  has  the  same  chemical  intensity  as  that 
in  the  body. 

That  such  gas  mixtures  are  to  be  found,  at  least 
in  all  cases  of  very  small  concentrations,  is  apparent. 
But  also  in  cases  with  greater  concentrations  equation 
(21)  has  shown  itself  to  be  applicable  in  this  way,  that 
we  assume  in  the  substance  other  constituents  than 
those  which  are  generally  considered  as  being  present, 
For  example,  in  a  salt  solution,  water  and  NaCl,  we 
assume  also  Na  and  Cl  as  constituents.  This  hypothesis, 
of  Arrhenius,  of  the  dissociation  into  ions  has  made  it 
possible  to  apply  formula  (21)  very  widely,  and  to 
make  it  of  great  value. 


124  PRINCIPLES  OF  MATHEMATICAL    CHEMISTRY. 

In  our  subject,  of  the  mathematical  consideration 
of  natural  processes,  the  grounds  for  or  against  such 
an  hypothesis  need  not  be  considered.  We  apply 
formula  (21)  to  non-gaseous  substances,  and  draw  the 
mathematical  consequences  of  the  assumption. 

The  single  condition  that  it  is  necessary  to  make  in 
the  property  of  the  function  77,  in  order  that  our  as- 
sumption may  be  upheld,  is  the  following: 

The  energy  equation  (i)  must  remain  true  when 
by  unchanged  6,  P,  LTl  .  .  .  77«,  all  masses  Mlt  M^  .  .  . 
Mn  are  proportionally  increased.  The  value  of  each 
intensity  must  not  change  (but  still  it  can  be  consid- 
ered as  a  function  of  0,  P,  Mlt  M^  .  .  .  Mn)  by  a  pro- 
portional increase  of  all  the  terms  M.  The  expression 
C  has,  according  to  9  (page  69),  this  property,  therefore 
II  has  it  also. 

The  other  properties  which  the  function  77  must 
show  in  the  constituents  of  non-gaseous  substances 
can  be  ascertained  by  the  method  which  led  to  equa- 
tion (2),  page  (80).  We  bring  each  of  the  equations  (2), 

dE  5  6dS  -  PdV  +  Tl.dM, 


into  the  form 
(23) 


Since  the  expression  on  the  left  is  a  complete  dif- 


PROPERTIES   OF  CHEMICAL   INTENSITY.       12$ 


ferential,  it  follows  that,  for  all  reversible  processes, 
have  we  the  equations 


(24) 


_L  =  _   3S_       977,  _ 
90  "        9^'      9^  ~ 
977,  _  977,  977,  _ 


977,  =  _ _  _95        977,^ 
90  "        9J/,'      dP 
977,  __  977,  977,  _ 


If  we  substitute  the  quantities  v  and  s  which  are,  in 
general,  functions  of  the  pressure,  the  temperatures, 
and  the  masses, 


then 
(26) 


The  equations  which  show  the  relations  between 
the  intensities  and  the  masses,  lead  to  a  consequence 
which  is  worthy  of  notice.  If  U1  depends  besides  on 
Pand  0,  only  on  the  concentration,  then 

(27)  Ct  =  Nl:(tfl  +  N.  +  .. 


Jf.\ 


mj 


m^\m^       mt 
and  77?  in  the  same  way  depends  on  C^  etc. 


126  PRINCIPLES  OF  MATHEMATICAL    CHEMISTRY. 

Here  each  TV  means  the  number  of  mols,  w,  of  one 
of  the  n  constituents,  which  are  present  in  the  phase 
considered.  Now 


.       ..      ,       ,        _ 

*       "  "" 


_,    ^,,     *  , 

'  mr  m  '  m' 


and  the  equations  under  (24)  require  that 


or,  in  general,  that  the  expression 

(38)  |g.ClW 

has  the  same  value  for  all  constituents. 

For  gaseous  constituents  this  value,  by  (20),  is  equal 
to  R06\  therefore  we  haver/0r  all  constituents, 


_ 

dC~  mC~~C1 
or 


which  is  the  same  as  (21),  where  only  one  of  the  con- 
stituents  is   gaseous,  to   which   the   concentration    0 


PROPERTIES   OF  CHEMICAL   INTENSITY. 

refers ;  or  where  only  one  constituent  is  present  which, 
by  the  same  pressure  and  temperature  in  the  gas  mix- 
ture, can  be  in  equilibrium  with  the  substance. 

The  general  utility  of  equation  (21)  is  made  de- 
pendent upon  the  assumption  that  the  chemical  in- 
tensity of  each  constituent  of  a  body  is  a  function 
only  of  its  pressure,  temperature,  and  concentration 
in  the  body.  We  have  up  to  the  present  left  an 
arbitrary  choice  of  the  independent  constituents. 
Nothing  prevents  us  from  considering  these  (which 
have  to  fulfil  no  other  condition  than  of  being  in- 
dependent) as  compounds  themselves,  and  made  up 
of  other  elementary  constituents,  as,  e.g.,  of  their  ions. 
We  will  now  assume  that  it  is  possible  so  to  define  a 
constituent  that  the  intensity  of  each  one  depends  (besides 
upon  its  temperature  and  pressure)  upon  its  concentra- 
tion alone.  Then  formula  (21)  holds.  By  the  ionic  hy- 
pothesis of  Arrhenius,  such  a  choice  of  the  constituents, 
in  solutions  of  electrolytes,  is  possible. 

Just  as  we  explain  the  apparent  deviations  from 
Avogadro's  law  by  the  proper  assumption  of  the  mo- 
lecular decomposition  into  constituents,  so  in  this  case 
we  succeed  in  bringing  the  facts  under  the  general  law 
by  a  proper  choice  of  the  chemical  constituents  of  the 
phenomena.  We  look  upon  the  hypothesis  of  Arrhe- 
nius, therefore,  as  a  fitting  definition  of  the  concep- 
tion of  "  chemical  constituents." 

To  show  the  application  of  equation  (21),  we  will 
give  the  E.  M.  F.  which  exists  in  two  places  in  a  liquid, 
which  in  consequence  of  different  concentrations,  C^  and 
£T?,  show  a  difference,  77^  —  JIa,  in  their  chemical  intensi- 


128   PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

ties,  but  which  are  at  the  same  temperature  and  pres- 
sure.    From  equation  (n)  (page  106)  we  obtain 


(29) 


If  the  valence  of  the  liquid  is  wt  then  the  equivalent 
a  equals  m  :  w,  where  m  is  the  molecular  weight,  and 
we  have 


The  derivation  by  Gibbs  of  the  law  of  absorption  is 
another  application  of  equation  (21).  If  a  gas  stands 
over  a  liquid  which  absorbs  it,  then  when  equilibrium 
is  reached  the  intensity  of  the  pure  gas  is  the  same  as 
that  of  the  absorbed  one,  that  is,  when  the  pressure 
and  temperature  of  the  two  phases  have  become  equal- 
ized. If  we  designate  by  C0  and  C0f  the  concentrations  ; 
by  m^  and  m{  the  molecular  weights,  of  the  solvent  in 
the  liquid  and  gaseous  phases;  by  Cv  and  C/  the  con- 
centrations ;  by  m1  and  *#/  the  molecular  weights,  of 
the  gases  in  the  solution,  and  in  the  vapor  space  over 
it;  and  if  further  A0,  A0',  A19  and  A/  are  functions  of 
the  pressure  and  temperature,  then 

A.  +  m0R0ic0  =  A:  +  m;R;ic;< 

A,  +  m.RJC,  =  A,'  +  mfRJCf. 

It  follows,  especially  for  the  case  that  the  vapor  of 
the  liquid  possesses  a  negligible  partial  pressure  as 


PROPERTIES   Of    CHEMICAL   INTENSITY.        129 

against  the  gas  in  the  vapor,  i.e.,  the  latter  has  the 
concentration  C{  =  I, 


Ct  =  F(9,  P). 

That  this  function,  F,  of  the  pressure  and  tempera- 
ture is  for  small  absorption,  i.e.,  small  value  for  C}  y 
almost  independent  of  the  pressure,  as  Henry  found,  is 
established  by  (26)  (page  125).  For  small  ClfAl  is 
large  in  comparison  to  A/,  and  Al  varies,  according  to 
that  formula,  very  slightly  with  the  pressure. 


CHAPTER   II. 

THE   SIMPLE   CHEMICAL  REACTION. 
WE  turn  again  to  equation  (i)  on  page  114, 
(i)  dE 5  OdS -  PdV+  JI.dM,  +  IIJM,  +  ... IIndMn, 

which  holds  for  any  substance  that  can  take  up  or  give 
out  heat  or  volume  energy,  and  which  possesses  n  con 
stituents,  whose  amounts  can  change  independently  of 
one  another.  We  will  now,  however,  consider  a  special 
kind  of  change  between  the  constituents  of  the  mixture, 
namely,  a  chemical  reaction,  whose  special  property  is 
the  existence  of  a  stoichiometrical  dependence  between 
the  n  changes.  By  the  decomposition  of  water,  for  ex- 
ample, we  consider  three  independent  constituents  be- 
side one  another  ;  they  are  water,  oxygen,  and  hydrogen. 
These  form,  in  each  instant  of  the  progress  of  the  re- 
action, a  mixture  for  which  equation  (i)  holds;  they 
can,  by  other  influences,  increase  or  decrease  independ- 
ently, but  the  chentical  reaction  restricts  their  freedom, 
in  such  a  way  that  when  we  know  the  change  of  one  of 
the  three,  which  takes  place  during  the  process,  we  can 
also  give  the  change  of  each  of  the  other  two. 

We  call  the  chemical  reaction  simple  when,  as  in 

330 


THE   SIMPLE    CHEMICAL   REACTION.  131 

this  example,  the  changes  of  mass  which  take  place  at 
the  same  time  are  related  so  that  from  the  change  of 
one  mass  we  can  find  the  changes  for  the  others.  If 
the  molecular  weights  of  the  substances,  whose  masses 
are  Mlt  M9,  .  .  .  Mn,  are  mlt  mt,  .  .  .  mn,  and  if,  in  con- 
sequence of  the  stoichiometrical  relations,  vlt  v±,  ...  vw 
mols  of  the  substances  enter  with  each  other  into 
chemical  reactions,  then  we  can  place 

(2)dMl=vlmldM0  ,     dMf=vjnjlM^  .  .  .  dMn=vnmndMn. 

The  terms  v  we  will  call  the  "  exchange  numbers  " 
(Umsatzzahlen)  of  the  single  substances  in  the  reaction. 
dMQ  is  a  factor  ;  if  hydrogen  takes  part  in  the  reaction 
with  the  "exchange  number"  I,  then  dMQ  is  the 
amount  of  H  formed  in  an  infinitely  small  element  of 
time  in  the  reaction.  On  the  other  hand  we  can  imagine 
anyone  of  the  substances  formed  placed  in  a  simple 
reaction  with  hydrogen  with  the  "  exchange  number  " 
I.  In  the  case  of  the  decomposition  of  water,  for  ex- 
ample, we  have,  when  we  distinguish  water,  oxygen, 
and  hydrogen  by  the  indices  i,  2,  and  3, 

m,  —  1  8,      ra,  =  32,       mz  =  2. 
'  vl  =  —  2,     v^  =  +  i,     z/8  =  +  2. 

By  this  the  stoichiometrical  relation 

HaO  =  0  +  H2,     or     2H20  =  03  +  2H9 
is  placed  in  (2)  in  the  form 


and  the  chemical  meaning  of  dM0  is  made  clear. 


132   PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

We  will  designate  by  M0  ,  the  variable  upon  which 
alone  the  reaction  depends,  as  the  parameter  of  the 
reaction. 

The  addition  of  equations  (2)  gives,  when  the  total 
mass  is  equal  to  M, 


(2b)  dM  = 


where  M  can  be  placed  instead  of  M0. 

By  the  substitution  of  the  values  in  (2)  in  (i)  we 
have 

(3)    dE>6dS-PdV 

+  \'nivlml  +  n,v,m,+.  .  .  IInvnmn\dM,. 

The  factor  of  dM0  ,  which  we  will  designate  by  $, 
can  be  calculated  when  all  reacting  constituents  are 
perfect  gases  and  form  a  homogeneous  gas  mixture. 
We  obtain  from  equation  (15),  page  12  ir 


(4)         #>fo-&l  +  ^10)^1 

+  (^-^9  +  ^K^9 

and  from  equations  (18)  and  (19),  page  122, 

(5)      #  >  E.  +  cpo  -  s0"e  -  cpeie  +  R 


f*  .  .  .  A".). 


THE  SIMPLE   CHEMICAL  REACTION.  133 

Here  we  use,  according  to  (6),  page  68, 

R0  =  m.R,  =  mzRz  =  .  .  .  =  2  cals. 
and 


VjKj&i  +  WfKjRi  +  •  .  ., 


(6), 


E0  is  therefore  the  energy  and  S0rr  the  entropy  of  I 
mol  of  the  reacting  gas  mixture  for  a  certain  standard 
state.  C*  is  the  molecular  heat  of  the  mixture  for 
constant  pressure,  and  N  the  sum  of  the  "  exchange 
numbers."  Since  the  total  change  of  mass  is 

(7)  dM  =  (m.v,  +  m^  +  .  .  .  mnvn)dMQ  , 

then  by  (3) 

(8) 

ll  ^       .  .  .     nn 

which  we  can  designate  as  the  chemical  intensity  of 
the  reaction. 

If  the  constituents  are  not  perfect  gases,  but  form  a 
homogeneous  mixture,  then  by  (21),  page  123, 


*  The  indices  will  distinguish  these  sufficiently  from  the  concen- 
trations. 


134  PRINCIPLES   Of  MATHEMATICAL    CHEMISTKY, 


where  0(#,  P)  is  a  function  of  the  temperature  and 
pressure,  which  is  the  value  that  the  expression 

(10)  #  =  n,m,v,  +  n,m,v,  +  .  .  .  nnmnvn 

assumes,  when  each  of  the  concentrations  C,  ,  C9t  Ca, 
.  .  .  Cn  is  equal  to  i. 

If  the  constituents  do  not  form  a  homogeneous 
mixture,  but  belong  to  different  phases,  then  we  have 
to  use  the  above  equation  for  each  phase  and  add  them 
together. 

We  will  now  apply  equation  (3)  to  a  case  of  chemi- 
cal equilibrium.  The  reaction,  we  will  assume,  is  fin- 
ished, then  changes  of  heat  and  volume  energy  do  not 
take  place,  the  energy  does  not  change  ;  but  a  contin- 
uance of  the  reaction  either  forward  or  backward  is 
still  possible,  as,  for  example,  a  change  in  temperature 
or  pressure  would  show. 

In  order  that  the  left  side  of  (3),  dE,  equals  o,  since 

9t  dS,  and  dV  have  the  same  value,  the  factor  of 
i.e.  <£,  must  have  the  same  sign  as  dM9  ,asdS=o 
and  dV=o  show.  If  dM0  can  be  chosen  positive  or 
negative,  i.e.,  if  the  reaction  is  reversible,  or  if  the 
chemical  reaction  can  continue  or  go  backwards,  ac 
cording  to  the  same  stoichiometrical  laws,  then 


^  4-  .  .  .  =  o  • 
or  when  A  is  a  constant, 


THE  SIMPLE   CHEMICAL  REACTION.  135 

in  the  case  that  all  reacting  gases  are  perfect ;   other- 
wise 

a  function  of  pressure  and  temperature  of  which  we 
know  nothing  further. 

Before  we  show  the  application  of  these  important 
formulae,  we  must  observe  that  by  the  differentiation 
of  (ii 


by  constant  pressure. 

If  we  substitute  C,  +  RQN  for  Cp,  where  Cv  is  the 
molecular  heat  of  the  mixture  at  constant  volume,  then 
by  a  simple  operation  we  obtain 


«> 


A  glance  at  formula  (6)  shows  that  £0  +  CJB  is 
the  intrinsic  energy  gained  by  the  mixture,  when  the 
reaction  has  gone  so  far  as  to  form  I  mol  ;  therefore 
the  negative  heat  of  reaction  for  constant  volume. 
Further,  we  see  that  E0  +  CP0  is  the  intrinsic  energy 
increased  by  the  work  of  expansion,  i.e.,  the  negative 
heat  of  reaction  for  constant  pressure.  If  by  Q  we 
designate  the  negative  heat  of  reaction  by  constant 
pressure,  then 

(14)         Q  =  +  R.p*j(c*ct-  .  .  .  £,*). 


136  PRINCIPLES  OF  MATHEMATICAL    CHEMISTRY. 
It  also  follows  from  (nb)  that 

Q  =  +  X.P(PW  •  •  •  P.-), 


and  corresponding  equation's  are  obtained  from  (13)  for 
the  negative  heat  of  reaction  for  constant  volume. 
That  equations  (7)  and  (8),  pages  81,  82,  are  special 
cases  of  these  equations  is  apparent. 

Equation  (14)  can  be  derived  in  another  way,  not 
only  restricted  to  gases.     We  have 


which  for  constant  pressure  and  temperature  goes  over 
into 


i.e.,  according  to  (24),  page  125, 


(15)  dS=, 


and  the  necessary  supply  of  heat  is 


If  now  the  differential  quotient  —  —  is  independent 

QV 


THE  SIMPLE   CHEMICAL  REACTION.  137 

of   the  mass,  then  by  integration,  as  on  pages  8l,  82, 
we  have 

(15.) 

Between  equations  (8),  (9),  and  (10),  however,  we 
have  the  relation 

(16)       n  =  n.(6,  p)  +  Rei(c*ct-  .  .  .  cm~)  •, 

and  when,  in  consequence  of  the  equation 


we  place 

(17)  R,  =  R(mj>i  +  m%v%  +  .  .  .  +  mnvn\ 

and  designate  by  II0  the  value  of  IT  for  Ct  =  C9=  .  .  .  =  i, 
and  partially  differentiate  (16)  with  respect  to  6,  then 

Oll0 


Since  in  case  of  equilibrium  H  =  o,  then 


It  follows  when  the  concentrations  C  are  assumed  as 
determining  the  equilibrium,  therefore  dependent  upon 
6,  that 


*  The  Q  used  in  this  case  is  the  negative  heat,  while  on  page  81  it 
is  the  positive  heat,  so  that  the  two  equations.  are,  precisely  the  same.  — 


TKANS. 

UNIVERSITY 


138   PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

hence 

(18)  AQ  =  +  RPCW  .  .  .  C: 


The  amount  of  heat  which  is  to  be  supplied  for  the 
formation  of  i  mol 

AM  =  m  =  #2,2^  +  mjUi  +  .  «  .  #V« 
is  therefore,  in  accordance  with  (17), 

(18*)  Q  =  +  R.P  J0  l(C:>C*>  .  .  .  €.''), 

which  is  the  same  as  (14),  derived  for  gases  alone- 

This  relation,  as  van't  Hoff  has  already  shown,  means 

that  by  increasing  temperature  each  chemical  reaction 

that  absorbs  heat  goes  forward,  and  each  one  that  de- 

velops heat  goes  backward. 

The  supply  of  heat  at  constant  volume  we  find,  ac- 

cording to  a  rule  given  in  a  previous  chapter,  from  (13), 

as 

(19)  Q  =  Rp  £  i(c^c;>  .  .  .  67«)  -  Rjre, 

where  N'  mols  of  gas  are  formed  by  the  reaction. 

By  the  integration  of  equations  (18$)  and  (19)  it 
is  to  be  remembered  that  the  heat  of  reaction  depends 
upon  the  temperature  of  the  same  (see  page  18). 

Each  of  the  two  derivations  of  equation  (13)  can 
be  used,  when  correspondingly  altered,  to  develop  the 
equation 

(20)  F  =  -  Ro 


THE   SIMPLE   CHEMICAL  REACTION*.  139 

in  which  F0  (=NRQ6/P)  is  the  volume  of  as  many  mols  in 
gaseous  state,  as  the  sum  TVof  the  "  exchange  numbers  " 
is  equal  to.  The  formula  follows  in  the  same  way  as  (  14) 
by  differentiation  of  (11),  and  also,  in  general,  for  non- 
gaseous  bodies,  by  aid  of  (24)  (page  125),  namely, 


There  is  one  consequence  of  (14)  that  is  worthy  of 
notice.  If  a  reversible  reaction  proceeds  without  de- 
velopment of  heat,  then  we  can  only  conclude  that  the 
product  of  the  concentrations  C^C^  .  .  .  CJn,  for  the 
temperature  of  the  reaction,  is  a  maximum  or  a  mini- 
mum, and  the  same  holds  true  for  (20).  When,  how- 
ever, by  the  change  of  the  temperature  of  reaction, 
there  is  no  heat  developed,  then  the  concentration 
product  is  independent  of  the  temperature.  Also 
when  by  change  of  pressure  no  change  of  volume 
follows,  then  the  concentration  product  is  also  inde- 
pendent of  the  pressure,  i.e.,  in  general,  a  constant.  If 
this  holds  for  greater  concentrations,  i.e.,  independ- 
ent of  the  concentrations  C,  then  it  is  only  possible 
that  these  cancel  one  another,  i.e.,  the  concentration 
product  is  I  and  its  logarithm  is  o.  In  other  words, 
then,  no  reaction  takes  place;  the  end  product  is  the 
same  as  the  initial  ;  to  each  C+v  there  is  a  C~v. 

Since,  by  the  phenomenon  observed  by  Hess 
of  the  thermo-neutrality  of  dilute  salt  solutions,  in 
many  cases  no  heat  is  observed  by  mixing,  we  are  led 
to  the  assumption  that  in  these  cases  there  is  no  reac- 


14°  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

tion,  i.e.,  that  such  salt  solutions  are  composed  of  sub- 
stances that  are  not  changed.  These  constituents  we 
find  to  be  the  ions  ;  the  dissolved  salts  are  dissociated 
into  their  ions,  when  we  observe  no  heat  development 
by  the  mixing  of  their  solutions. 


CHAPTER  III. 

CHEMICAL  EQUILIBRIUM. 

ACCORDING  to  equation  (12)  (page  135)  a  reversible 
chemical  reaction  can  only  come  to  rest,  i.e.,  be  in  equi- 
librium, when  the  product  of  the  concentrations  and 
exchange  numbers  of  the  reacting  constituents, 

(i)  C7-C°  •••£,*", 

possesses  a  certain  value,  which  depends  upon  the  tern- 
perature  and  pressure. 

The  concentration  of  a  constituent,  in  a  homo- 
geneous mixture,  is  the  ratio  of  the  number  N  of  its 
mols,  to  the  total  number,  N^  +  N^  +  .  .  .  +  N*  >  m 
phase  (not  in  the  system)  : 


according  to  (9)  (page  69).     Since  now  the  sum  of  all 
the  mols  does  not  change,  the  product, 

(2)  N?*N?...Nnv; 

is  a  quantity  which  is  only  dependent  upon  the  tem- 
perature and  pressure.    The  quantities  N,  which  change 

11 


142   PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 

by  continued  reaction,  are  the  amounts  of  the  reacting 
constituents  measured  in  mols.  Each  constituent  is 
therefore  to  be  measured  by  another  unit,  its  mol. 

If  all  reacting  constituents  in  the  system  are  gases, 
then  we  can  assume  (equation  (i  \b\  page  134)  that  the 
product, 

(3)  /VW'.  ..//«, 

i.e.,  the  partial  pressures  raised  to  the  powers  of  the 
exchange  numbers,  must  be  independent  of  the  pres- 
sures, and  dependent  only  upon  the  temperature. 

Since  during  a  reaction  the  amounts  of  single  con- 
stituents increase,  and  others  decrease,  the  exchange 
numbers  of  some  are  positive  and  of  others  are  nega- 
tive. If  now  va,  vb  .  .  .  are  the  absolute  values  of  the 
former,  and  vat  vp.  .  .  the  absolute  values  of  the  latter, 
then  e^ach  of  the  characteristic  products  (2)  and  (3)  can 
be  written  as  the  quotient  of  two  such  products;  for 
example,  the  first  assumes  the  form 


(4) 


In  the  numerator  is  the  number  of  increasing  mols, 
and  in  the  denominator  the  number  of  decreasing  ones. 

This  expression  assumes  a  very  convenient  form 
when  we  measure  the  constituents  in  reaction  equiva- 
lents vm,  instead  of  mols  ;  so  that  of  all  constituents, 
in  the  same  time,  an  equal  number  of  these  units  is 
transformed. 

If  at  first  we  have  A,  B,.  .  .  equivalents  of  the  in- 


CHEMICAL   EQUILIBRIUM.  143 

creasing  constituents,  and  A,  B,  .  .  .  equivalents  of  the 
decreasing  ones,  then  after  the  reaction  we  have  A  -\-  x, 
B  -f-  x,  .  .  .  of  the  one,  and  A  —  x,  B  —  x  ,  .  .  .  of  the 
other,  when  x  is  the  number  of  equivalents  transformed. 
If,  for  simplicity,  we  further  designate  the  absolute 
values  of  the  exchange  numbers  by  a,  b,  .  .  .  and  a, 
ft,  .  .  .  ,  then  for  the  case  of  equilibrium  we  have 

(5)  (A+  x)'  (B  +  *)>...  =  K(A  -  *Y  (B  -*?>.., 

where  K  is  a  function  of  pressure,  and  temperature. 
The  determination  of  chemical  equilibrium  is  there- 
fore in  all  cases  dependent  only  upon  the  solution  of 
an  algebraic  equation.  In  this  form  the  result  of  the 
last  chapter  was  found  by  Guldberg  and  Waage  in 
1867,  independently  of  the  principles  which  we  have 
used.  Since  then  it  has  been  proven  for  many 
important  cases  ;  we  will  now  treat  a  few  typical 
examples- 

For  the  formation  and  decomposition  of  hydriodic 
acid  according  to  the  formula 

H.+  I.  =  2HI, 

when  HI,  H,  and  I  are  distinguished  by  the  indices  i, 
2,  and  3,  we  must  substitute  in  (3) 

z>,  =  2,     vt  =  —  i,     v»  ~  —  i. 
We  obtain 


-<; 

AA 


144  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

where  K  is  a  function  of  the  temperature  independent 
of  the  pressure.  If,  at  first,  in  one  volume  we  have  so 
much  hydrogen  and  iodine,  that  their  partial  pressures 
are///  and//,  and  the  ratio  of  mixing  is 

(7)  *  =//:///, 

then  after  a  very  gradual  reaction,  when  the  reaction  is 
at  rest,  a  part  of  each  of  the  substances  has  been  used 
up  for  the  formation  of  hydriodic  acid,  they  are,  from 
stoichiometrical  grounds,  equal  to 

(8)  A/=A+4''    A  =  A  +  4- 

Zi  2 

If  x  is  the  ratio  of  the  H  still  free,  to  that  amount 
originally  introduced,  then 


(9) 


a 


and  if,  further,  P  is  the  total  pressure  of  the  mixture 
when  in  chemical  equilibrium,  then 


(10)  -P= 

and  it  follows  that 

-*),    A  -•/*•*» 


and  equation  (6)  becomes 


CHEMICAL  EQUILIBRIUM.  145 

If  at  first  equal  amounts  of  H  and  I  were  intro- 
duced, or  if  at  first  only  HI  was  introduced,  then  we 
can,  according  to  Lemoine's  experiments,  take^r  =  0.28 
approximately  (Nernst).  By  this  K  =  4  .  0.72'  :o.282 

=  26.45,  an<3  by  calculation  we  find —  or  26.0+.     If 

j 

the  state  of  equilibrium  is  observed  for  one  mixture, 
then  it  is  known  for  all.     By  aid  of  (7)  we  can  write 

(12)  in  the  form 

4(1  -*)'  =  ,&(*- i +  *). 

(13)  (K  -  4K  +  Kxz  -  (K  -  8)*  -  4  =  o. 

By  this  for  each  z  we  can  find  the  corresponding  x, 
by  solving  an  equation  of  the  second  degree,  as  Nernst 
has  done.  He  finds,  for  example, 

z  =  i.ooo        0.784        0.527        0.258 
*=  0.280        0.373         0.534        0.754- 

while  Lemoine  observed 

x  =  0.240  to  0.290  0.350  0.547  0.774 

It  is  simpler,  however,  to  calculate  the  values  of  z 
for  a  few  of  x  and  then  to  interpolate  the  others.  The 
graphic  method,  though  little  used  in  chemistry,  is  the 
one  best  adapted  for  a  complete  view  of  a  process,  and 
for  the  accuracy  of  observation.  If  x  and  z  are  con- 
sidered as  coordinates,  then  equation  (13)  gives  a 
curve  of  the  second  order.  If  we  transpose  the  x  axis 
so  that 

;,*£>* 

„=,+ 


146  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 
then  the  equation  becomes 
(14)  (A"  -4X  +  ***'-  4  =  0, 

which  is  a  hyperbola,  whose  asymptotes  are 

*'  K- 


which  intersects  the  straight  line  z'  =  o  in  the  two 
points,  x  =  ±  2  :  VAT—  4.  It  goes  also  throu'gh  the 
point  x  —  i,  ^  =  o,  and  meets  the  x  axis  in  the  two 
points  of  intersection  x  =  (K  —  8  ±  K)  :  2(K  —  4),  i.e., 
in  the  point  x  =  i  and  in  the  .point  x  =  —  4  :  (K  —  4). 
Using  as  the  value  for  K  the  one  chosen  by  Nerrist 
(as  given  above),  we  obtain  the  curve  given  in  Fig.  6. 


FIG.  6. 


If  we  place pH~.  i,  then  all  the  pressures  will  be 
represented  in  the  figure  by  distances, 


CHEMICAL   EQUILIBRIUM. 


/,  =  2(1-*),     f.=X,     A=*~(l-x). 

In  the  example  just  treated,  the  gas  mixture  does 
not  change  its  pressure,  when  the  reaction  follows  by 
constant  volume,  nor  its  volume,  when  by  constant 
pressure,  so  long  as  the  temperature  remains  the  same; 
for  from  each  two  volumes  (or  2  mols)  of  the  elementary 
constituents,  two  volumes  of  hydrogen  iodide  are 
formed.  In  general,  however,  by  the  dissociation  of  a 
gas  into  its  gaseous  constituents,  as  also  by  the  reverse 
process,  the  volume  changes.  Thus  by  the  dissocia- 
tion of  nitrogen  dioxide 


of  phosphorous  pentachloride, 

PC16  =  PC13+C12; 

of  methylether  hydrochloride  in  hydrochloric  acid  and 
methylether, 

(CH3)aO.HCl  =  (CH8XO  +  HC1  ; 
or  of  carbon  dioxide, 

2COa  =  Oa  +  2CO, 
etc. 

When  from  TV  mols  of  a  substance,  N0  dissociate, 
each  into  n  mols,  then  we  have 

N-  N0  +  nN,  =  N+  (n  -  i)N0  mols  present. 
If  by  this  the  volume  is  held  unchanged,  the  pressure 
PO  of  the  original  undissociated  substance  becomes  P 


148   PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

in  the  mixture  which  results  from  the  dissociation. 
According  to  equation  (7)  (page  68),  at  temperature  6 
we  have 

P9V=  NRQ0,     PV  =  \N  +(n  -  i)N0]R,0. 

P 

(16)  p  =  l  +  ("  ~  0?» 

*4i 

in  which  the  ratio 

(17)  rf  =  N0:N 

is  designated  as  the  degree  of  dissociation.  If  we  now  call 
the  partial  pressure  exerted  by  the  undissociated  por- 
tion of  the  original  substance  /,  and  that  exerted  by 
each  of  the  n  products  of  dissociation  which  arise  from 
each  molecule /',  then 

(18)  P.=f  +  S; 

(19)  />  =  /  +  «/; 


and  it  follows  from  (16)  that 
(21)  r,  = 


If  by  the  dissociation  instead  of  holding  the  volume 
constant,  we  keep  the  pressure  so,  at  P0  then  the  volupne 


CHEMICAL  EQUILIBRIUM.  1 49 

changes  from  F0  to  F,  and  it  follows  from  the  equa- 
tions 

/> F0  =  IfR.O,    PaV  =  [(N  +  (n  -  i)N^R.e  ; 

that  for  the  degree  of  dissociation  we  have  the  rela- 
tion 


where  30  and  3  represent  the  densities  before  and 
after  the  dissociation,  measured,  for  example,  with  re- 
spect to  atmospheric  air.  Further,  we  find 

*  =  (jr^' 

jj r>  7^3    —   30  r>  7^3    —   30 

f     — -    •*-~~, T~T"    —     *    n7  ^T    > 


We  can  also  place,  by  taking  3'  as  the  density  by 
complete  dissociation,  i.e.,  when  77  =  i,  or 

3'  =  30  :  », 
3  -  3'  3-3' 


From  the  equations 

pv  =  RM6,    p'V=R'Med 

we  can  also  find  the  amount  M  of  the  substance,  still 
remaining  in  the  original  state,  and  the  amount  M'  of 
each  of  the  n  equal  parts  which  arise  from  the  dissocia- 


PRINCIPLES  OF  MATHEMATICAL    CHEMISTRY. 


tion.  R  is  the  specific  gas  constant  of  the  undisso- 
ciated  substance,  and  R'  is  that  of  one  of  the  products 
of  the  dissociation.  We  find 


R0(n  - 


RO    (n  -  i)30' 


According  to  the  law  of  dissociation,  at  the  entrance 
of  equilibrium,  expression  (3)  must  equal  a  quantity  K, 
which  depends  only  upon  the  temperature  ;  thus  when 
all  the  molecules  are  of  the  same  kind  we  have 


(23)  («/)»/-'  =  /if,    p'n  =  K'.p, 

where  K'  =  K\  nn  is  a  new  constant.  The  dependence 
between  p'  and  /  can  therefore  be  shown  by  a  higher 
parabola,  or  in  the  case  n  —  2  by  an  ordinary  one  (Fig. 
7).  The  dependence  of  P  and  P  can  then  also  be  seen. 


Gibbs  first  proved  this  theory,  in  that  he  showed 
by  decomposition  of  nitrogen  dioxide,  N2O4,  into  two 
molecules  of  NO,,  that  the  expression 

(3,  -  a)' 


log  /-(2/)°  = 


.  2 


CHEMICAL   EQUILIBRIUM.  I$I 

was  the  same  as  the  temperature  function  in  equation 
(11),  page  134,  and  this  expression  he  calculated  from 
observations  by  Deville  and  Troost. 

The  connection  between  P  and  p  can  also  be  shown 
very  clearly  in  the  following  manner.  From  (19)  and 
(23)  it  follows 


(24)  (P-p)*  =  K.p. 

We  recognize  easily  by  the  ordinary  methods  that 
(24)  represents  a  curve  that  runs,  as  is  shown  by  Fig.  8, 


FIG.  8. 


at  point/  =  o  in  direction  of  P,  and  then  gradually  in- 
clines so  that  for  p  =  oo  it  is  at  an  angle  of  less  than 


152   PRINCIPLES  OF   MATHEMATICAL    CHEMISTRY. 

45°  to  the  /  axis.  In  the  case  n  =  2  it  is  an  ordinary 
parabola,  whose  axis  is  inclined  at  an  angle  of  less  than 
45°  to  the/  axis.  In  this  case  we  can  also  find  rj  from 
the  figure.  In  equation  (22), 

/  =  />(!  -17):  (I  +  17), 

rj  is  the  tangent  of  the  angle  Tt  which  the  radius  of  the 
point  (p  .  P)  forms  with  the  direction  of  the  parabola 
axis. 

The  dependence  of  the  degree  of  dissociation  t? 
upon  the  temperature,  is  given  by  substitution  of  (22), 
and  (22b)  in  (23)  : 


When  we  differentiate  the  logarithms  of  both  sides 
we  find 


Since  rj  Is  a  pure  fraction,  -     has  always  the  sign 

o" 

of  the  first  iactor  of  the  right  side,  and  the  degree  of 
dissociation  rises  and  falls  with  the  constant  K,  whose 
dependence  upon  the  temperature  is  apparent  from 
equations  ( u^)  and  (14$),  pages  134  and  136.  If  K 
rises  with  the  temperature,  and  if  the  dissociation  be- 
tween the  temperatures  #,  and  #a  is  nearly  complete, 
then  the  relation  which  exists  between  rj  and  6  has  the 
form  given  in  Fig.  9. 


CHEMICAL  EQUILIBRIUM. 


153 


The  course  of   the  dissociation  of  carbon  dioxide 
into  oxygen  and  carbon  monoxide, 

2CO2  =  2CO  +  O2, 

has  been  investigated  by  Le  Chatelier.     The  exchange 
numbers  (umsatz  zahlen)  of  the  three  constituents  are 


FIG.  9. 

and  for  the  partial  pressures,  which  we  will  designate 
A»A>  we 


where  P  is  the  total  pressure. 

Further,  when  only  CO2  is  present  at  first 

A  =  2A- 

Since  here  2  molecules  fall  into  3,  for  use  of  equa- 
tions (16)  to  (23)  we  must  place 


And   the  degree  of  dissociation  is   given  by  help   of 
equation  (21)  as: 


154   PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 
Further,  we  find 


By  this  equation  (n£),  page  134,  goes  over  into  the 
form 


-  r)      2        r,2 


If  we  now  calculate  a  table  giving  for  the  values 
77  =  o  to  77  =  i,  the  logarithms,  and  conversely  for 
each  value  of  the  latter,  the  corresponding  value  7;, 
then  for  each  pressure  P,  and  each  temperature  9, 
the  degree  of  dissociation,  /;,  can  be  found.  We 
must  observe,  however,  (i)  the  specific  heats  of  the 
three  constituents  by  constant  pressure,  from  which  by 
equation  (6),  p.  133,  the  value  CP  follows;  (2)  the  heats 
of  reaction  Q,  for  any  determined  temperature  6,  by 
constant  pressure,  from  which,  according  to 


(see  derivation  of  equation  (14),  p.  135),  the  constant 
£0  is  obtained  ;  (3)  a  single  value  of  y,  which  is  given  by 
a  certain  6  and  P,  —  in  order  to  be  able  to  calculate  the 
constant  A.  The  values  chosen  by  Le  Chatelier  are, 
for  many  reasons,  to  be  objected  to  ;  but  here  we  have 
not  space  to  enter  into  this  interesting  subject  any  fur- 
ther, but  must  reserve  it  for  a  future  opportunity. 


CHEMICAL  EQUILIBRIUM.  1  55 

We  will  not  give  here  an  example  for  equation  (14) 
on  account  of  the  lack  of  experimental  data,  although 
wide  applications  have  been  made  of  it  in  technical 
subjects. 

If  an  electrolyte  dissociates  into  its  ions,  as  accord- 
ing to  the  hypothesis  of  Arrhenius  is  the  case  with 
solutions,  its  degree  of  dissociation  77  =  ^:^00  (see 
page  109)  gives  the  concentration  of  each  ion.  If  there 
are  n  mols  of  the  electrolyte  in  solution,  then  there  are 
rjn  mols  of  each  ion  present  and  (i  —  if)n  remains  unde- 
composed.  If  n0  is  the  number  of  mols  of  the  solvent, 
then,  since  the  exchange  numbers  (umsatz  zahlen)  of 
the  ions  are  -\-  i  and  that  of  the  electrolyte  —  I,  the 
expression 


must  depend  only  upon  the  temperature  and  pressure. 
If  the  solution  is  highly  diluted,  i.e.,  n0  very  large  as 
compared  to  n>  it  follows  that 


Finally,  we  can  place  — °  proportional  to  the  dilution 
n 

v,  i  e.,  the  volume  to  which  i  gram  or  i  mol  of  the  salt 
is  brought:     Or 


156  PRINCIPLES  OF   MATHEMATICAL    CHEMISTRY. 


From  this  the  term  77  can  be  found  by  solving  the 
equation 

rf  +  Krjv  —  Kv  —  O, 


or 


The  dependence  between  the  degree  01    dissocia- 
tion y  and  the  dilution  v  is  represented  by  a  hyper- 
bola which  in  the  origin  touches  the  jy  axis  (Fig.  10). 
y 


2:  K 


FIG.  10. 

The  formula  is  supported  by  numerous  investiga, 
tions,  e.g.,  by  the  following  series  from  van't  Hoff  and 
Reicher.  For  acetic  acid  at  19°. I,  rj  is  the  ratio  of  the 
molecular  electrical  conductivity,  to  that  at  infinite  di- 
lution,  i.e.,  by  complete  dissociation.  By  the  dilution 
of  a  gram  molecule  (mol)  of  acetic  acid  to 

9.2697    4-     1 6-    64-     256-     1024  X  9.2697 


CHEMICAL    EQUILIBRIUM.  I  57 

the  molecular  conductivity  referred  to  mercury  was 
4.69     9.38     18.6     35.9     67.4     122  X  icr7 

Since  at  infinite  dilution  the  molecular  conductivity 
is  335  X  io~7,  the  degrees  of  dissociation,  are  in  parts 
per  hundred, 

IOOT?  =  1.40     2.80     5.55     10.7.     20.  1     36.4. 

From  equation  (25)  the  values  of  log  K  follow  ;  they 
have  the  characteristic  —  5,  and  the  mantissas  are 

33i  337  342  335  329  341 

The  value  of  K  is  thus  actually  independent  of  rj 
and  v. 

If  we  take  instead  of  the  degree  of  dissociation  77, 
the  ratio  of  the  molecular  conductivity  //,  to  the  cor- 
responding value  /i^  .for  infinite  dilution,  anxl  then  in- 
stead of  ju  the  specific  conductivity  cr  =  jun,  i.e., 

JA  __  pn  _    o_ 
^       W      »n*>  ' 

the    equation    nrf  :  (i  —  rf)  =  K'nQ  goes  over  into  the 
form 


<ra  +  n^K'v  -  nJCjfji  =  o, 
or 

JCpJ'  =  K'n.if,(n  +  \n, 


The  relation  between  n  and  <r  is  also  represented  by 
a  parabola,  which  cuts  the  n  axis  at  the  origin  at  an 
angle,  whose  tangent  is  ju^.  In  fact  the  observations 
of  F.  Kolrausch  show  such  a  behavior  at  high  and 


158   PRINCIPLES    OF    MATHEMATICAL    CHEMISTRY. 

medium  dilutions.     That  the  formula  is  not  applicable 
to  concentrated  solutions  is  shown  by  its  derivation. 

If  a  solid  body  decomposes  by  heating  into  several 
gases,  then  we  must  apply  the  equations  on  pages  134 
and  136  to  the  solid  as  well  as  the  gaseous  phases. 
Since,  however,  the  concentration  of  the  solid  is  I, 
the  gas  mixture  only  comes  into  further  considera- 
tion. Ammonium  carbimate,  for  example,  dissociates 
into  carbon  dioxide  and  ammonia  according  to  the 
formula 

NH4O  .  CO  .  NH2  =  CO,  +  2NH8. 

If/  is  the  partial  pressure  of  the  carbon  dioxide,/, 
that  of  the  ammonia,  then 


a  quantity  which  changes  only  with  the  temperature. 
If  the  total  pressure  is  produced  only  by  the  products 
of  the  decomposition,  whose  partial  pressures  in  this 
case  are  //  and  //,  then 


If,  however,  in  the  space  where  the  decomposition 
takes  place  there  is,  before  the  dissociation  begins,  be- 
side the  carbimate,  still  ammonia  or  carbon  dioxide, 
then  the  process  goes  on  until  the  entrance  of  equi- 
librium, which  fulfils,  as  before,  the  equation 


CHEMICAL  EQUILIBRIUM.  159 

where  /*  —  A  +A  agam  represents  the  total  pressure, 
which  is  only  dependent  upon  the  temperature.  This 
consequence  of  the  theory  has  been  proved  by  Isam- 
bert.  He  observed  at  34°.o  C.,  with  a  total  pressure  of 
17.0  mm.  mercury,  the  different  partial  pressures  by 
different  (initial)  excesses  of  ammonia,  or  of  carbon 


dioxide,  and  found  for       V/iA*<  instead  of  17.0,  the 
values 

17.0         16.5         16.7         1  8.1 

At  42°.  5  C.,  with  a  total  pressure  of  28.8  mm.,  he 
found  for  each  cubic  foot,  by  different  initial  states  the 
numbers 

28.9         28.4         28.6         29.2 

On  the  other  hand,  the  action  of  an  indifferent  gas 
added  to  the  system  was  entirely  different.  If  its  par- 
tial pressure  is  pt  ,  then,  by  formula  (3), 


i.e.,  it  is  without  influence  upon  the  ratio  of/t  and/,. 

If  n  mols  of  alcohol  are  mixed  with  I  mol  of  acetic 
acid,  acetic  ester  and  water  are  formed  by  a  very 
slow  reaction,  which  is  represented  by  the  equation 

CH8COOH  +  CaHBOH  "H  CH3COOH6Ca  +HaO. 

After  conclusion  of  the  reaction  all  four  substances 
are  present.  If  x  molecules  have  reacted,  and  if  at 
first  there  was  neither  water  nor  ethylacetate  present, 
then,  according  to  equation  (5), 


l6o  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

must  be  constant  in  value,  and,  according  to  observa- 
tion, equal  1/4.     From 

4(7*  —  x}(\  —  x)  =  x* 

it  follows,  since  to  the  value  n  —  o  the  value  x  =  o 
corresponds, 

x  —  \(n  +  i)  —  Vn*  —  n-\-  I. 
By  this  we  find 

for  n  —  0.05    0.20    0.50    i. o      2         4         8         5,00 
^  =  0.05    0.19    0.42    0.67    0.85    0.93    0.95     i. oo 

while  Berthelot  and  Pe~an  de  St.  Gilles  observed 

x  =  0.05    0.19    0.43    0.67    0.84    0.90    0.97    i.oo 

Further  examples  can  be  found  in : 

Ostwald,  Lehrbuch  der  allgem.  Chem.,  2d  ed.  vol. 
2,  1887. 

Nernst,  Theoretical  Chemistry.  Trans,  by  Dr. 
Palmer. 

Le  Chatelier,  Recherches  expe"rimentales  et  theo- 
riques  sur  les  e"quilibres  chimiques.  Paris,  1888.  Also 
in  Annales  des  mines,  1888. 

van't  Hoff,  Etudes  de  dynamique  chimique.  Am- 
sterdam, 1884. 


CHAPTER  IV. 

FREEZING  AND  BOILING    POINTS  ;  ALSO  VAPOR    PRES- 
SURES  OF  HIGHLY  DILUTED   SOLUTIONS. 

IN  a  closed  space  there  is  a  solution  and  its  vapor. 
If  n  mols  of  solvent  are  in  the  solution  and  «'  in 
vapor,  and  if,  further,  nl  mols  of  the  dissolved  substance, 
or  salt,  are  in  the  solution  and  »/  in  the  vapor,  then 
according  to  (eq.  9),  page  69,  the  concentrations  in  the 
liquid  phase  are  n\(n-\-  n^)  and  n^  :  (n  -J-  wj,  and  in  the 
gaseous  one  are  n' :  (n1  +  «/)  and  »,' :  (n'  -\-  ?//).  Accord- 
ing to  (eq.  21),  page  123,  the  chemical  intensities  are 


*+*,' 


n'+n," 

provided  that  only  reversible  processes  are  taken  into 
consideration.  Here  0  means  the  temperature  of  all 
parts  of  the  system,  the  quantities  with  stars  (*)  are  the 
intensities  that  would  be  shown  by  the  concentration 
I,  and  the  constant  R  is  related  to  the  gas  constant 
R0  =  2  cals  in  such  a  way  that 

(2)  mR  =  m'R'  =  m,Rv  =  m.'R,'  =  R^ 

where  m,  m',  m^  and  *»,'  are  the  molecular  weights. 

J6l 


1  62   PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

Now  by  the  law  proven  on  page  117  a  system  is  in 
equilibrium  when 

(3)  n  =  ir,  nt  =  n. 

The  pure  solvent  might  be  in  equilibrium  with  its 
vapor,  at  the  temperature  6,  when  its  pressure  is/0, 
while  the  solution  considered  might  be  in  the  same 
state  at  pressure  /.  Conversely,  by  this  pressure,  /,  the 
pure  solvent  would  not  boil  at  the  temperature  6,  but 
at  00. 

We  will  confine  our  further  treatment  to  highly 
diluted  solutions.  We  call  a  highly  diluted  solution  one 
in  which  the  chemical  intensity  of  the  solution  differs 
from  that  of  the  pure  solvent,  only  by  terms  which  con- 
tain the  first  powers  of  the  concentration  and  -temper- 
ature or  pressure.  If,  for  example,  we  develop  H  by 
Taylor's  series,  as  a  function  of  the  vapor  tension,  p, 
and  the  number,  #.  ,  of  mois  dissolved, 


(4)    77=7T.+^-A)  +  o«  +  higher  powers, 

where  the  terms  with  the  index  o  are  the  values  of  the 
intensities  3  and  their  differential  quotients,  for  /  =/0 
and«,=o;  then  for  highly  diluted  solutions  we  can 
neglect  the  higher  powers,  and  leave  the  equation  as 
above.  In  the  same  way, 

(43)       ,r  =  ;z:+(f)y-A)  +  (g>/. 

Since  now  the  pure  solvent  (nl  =  o),  at  the  pressure 
==/„,  is  in  equilibrium  with  its  vapor,  then 

' 


p 

(5) 


FREEZING   AND    BOILING   POINTS.  163 

and  we  obtain,  by  aid  of  (3), 


The  differential  quotients  on  the  left  side,  accord- 
ing to  (eq.  26),  page  125,  are  equal  to  the  specific  vol- 
umes, z/0,  #„',  of  the  pure  solvent  and  its  vapor  at  the 
pressure/  =/0  ;  those  on  the  right  follow  from  eq.  (i). 
We  obtain 


(7) 


If  the  salt  in  the  vapor  of  the  solution  is  unnotice- 
able  (i.e.,  n/  very  small  without  Rr  —  in  consequence  of 
strong  dissociation  of  the  solvent  by  evaporation  — 
being  very  large),  and  if,  further,  the  specific  volume  of 
the  liquid  is  negligibly  small,  as  compared  to  its  vapor, 
then  this  above  relation  becomes  simplified  to 


where  C  is  the  concentration  of  the  salt  in  the  solution. 
This  formula  (in  which  strictly  vj  should  be  sub- 
stituted by  equation  (7),  page  81,  has  been  proven  ex- 
tensively only  in  an  altered  form.  If  the  vapor  of  the 
pure  solvent  follows  the  equation  of  state  of  perfect 
gases,  then 

(9)  /.».'  =  R'O.- 

Since,  however,  R'  can  only  vary  from  R  when  the 


164  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

solvent  dissociates  by  evaporation,  which  possibility  we 
can  here  exclude,  we  can  place 


A 

In  this  equation  we  find  expressed  the  law  deduced 
experimentally  by  Raoult  in  1887,  which  reads:  A 
solution  boils  at  a  lower  pressure  than  the  pure  solvent  ; 
the  ratio  of  the  lowering  of  tJie  vapor  tension  to  the 
vapor  tension  of  the  pure  solvent,  the  so-called  "  relative" 
lowering  of  the  vapor  tension,  is  equal  to  the  concentration 
of  the  solution. 

Finally,  by  a  simple  alteration  of  equation  (10)  we 
can  derive  the  formula 


Since  /  does  not  differ  from  /0,  we  can  formulate 
the  law  as  follows:  The  "relative"  lowering  of  tne 
vapor  tension,  is  equal  to  the  ratio  of  the  number  of  mols 
dissolved,  to  the  number  of  mols  of  the  solvent. 

In  equation  (4)  the  intensity  was  treated  as  a  func- 
tion of  the  pressure  and  number  of  mols  dissolved. 
Since,  however,  the  pressure  of  a  saturated  vapor  de- 
pends oniy  upon  the  temperature,  we  can  also  look 
upon  the  intensity  as  a  function  of  the  temperature  and 
the  number  of  mols  dissolved,  and  place 


FREEZING  AND   BOILING  POINTS.  165 

By  aid  of  equations  (3)  and  (5)  it  follows,  further, 
that 

/air 


„       . 
"  a?      "  o)  *"  W*  '  ~       '<; 

and  according  to  equation  (26)  (page  125)  we  obtain 

r>n  J?f  ft 

(is)    W-0('-ft)  =     4^,+^..,', 

where  J0  and  J0'  represent  the  entropies  of  the  unit  of 
mass  of  the  pure  solvent  and  its  vapor.  We  substitute 
for  these  entropies  the  heat  of  evaporation  Q0  for  r 
mol,  m,  of  the  pure  solvent.  According  to  the  concep- 
tion of  entropy,  and  under  the  restriction  that  no  dis- 
sociation takes  place  by  evaporation, 

(14)  G*  =  ™0.(s.'  -  *o) 


where,  if  nl  is  very  small  as  compared  to  n,  we  can  place 


The  above  development  of  equation  (15)  is  for  the 
case  of  the  evaporation  of  a  solution.  But  all  the 
conclusions  hold  as  well,  for  the  case  that  a  phase 
separates  out  of  the  solution  in  any  other  form  than  a 
gaseous  one  ;  thus  for  freezing,  provided  only  that  the 
solid  body  separating  out  is  the  pure  solvent,  or  at 


1  66  PRINCIPLES   OF   MATHEMATICAL   CHEMISTRY. 

most  contains  a  negligible  amount  of  the  salt.  While 
go,  in  the  case  of  evaporation,  is  positive,  it  is  negative 
in  case  of  freezing., 

We  can  therefore  express  equation  (15)  in  words  as 
follows  :  A  solution  has  a  higher  boiling-point  and  lower 
freezing-point  than  the  pure  solvent;  this  temperature 
difference  (6  —  00)  is  proportional  to  the  concentration  of 
the  salt.  The  proportional  factor,  R0  #02  :  Q°0  =  f,  in- 
creases with  the  square  of  the  temperature  (#02),  and 
decreases  by  increasing  heat  of  aggregation.  The  value 
of  cr  is  generally  derived  from  observations  of  the  spe- 
cific increase  of  the  boiling-point,  or  from  the  specific 
decrease  of  the  freezing-point.  (The  specific  change  is 
the  change  of  temperature  shown  by  a  \<f>  solution,  i.e., 
where  mn  =  loo;/?//,).  ^  we  place  the  neat  °f  aggre- 
gation  as  referred  to  one  gram,  Q^\m  —  q^  and  Rjn^ 
=  R0,  then 


i  oom 


The  ml  multiple  of  this,  the  molecular  increase  of 
the  boiling-  or  decrease  of  the  freezing-point,  /*,  is  often 
given  in  place  of  cr, 

R06*       mf 

~~~  —      j 


100^0  '     100* 

The  laws  (10)  and  (15)  were  discovered  by  Raoult 
(1883)  and  van't  Hoff,  and  are  important  to  experi- 
mental chemistry,  as  a  means  of  finding  the  molecular 
weight  of  a  substance  in  solution,  which  often  differs 


FREEZING   AND    BOILING   POINTS. 

from  that  in  other  states.*    The  method  of  using  them 
will  be  apparent  from  the  following  examples  : 

According  to  the  results  of  Raoult,  11.346  grams 
of  oil  of  turpentine,  C10H]6,  molecular  weight  136,  dis- 
solved in  100  grams  of  ether  (C2H6),O,  molecular  weight 
74,  gives  a  solution  that  boils  at  a  pressure  of  36.01  cm. 
of  mercury,  while  pure  ether  shows  a  vapor  pressure  of 
38.30  cm.  It  follows,  therefore,  that  the  relative  lower- 
ing of  the  vapor  pressure  (left  side  of  equation  (10) 


38.30 


=  0.0598. 


On   the   other    hand,   we    know   that    11.346:136 
=  0.0834  mols  are  dissolved  in  100:74—  1.351.     The 

concentration  is  therefore  {-  —  r1  —  ) 

\n  -j-  nj 


l         =         82> 
1-434 


If  the  molecular  weight  is  to  be  derived  from  the 
observation,  then  we  must  find  the  number  of  mols  dis- 
solved from  equation  (io£), 

2.29 
-—.,.351=0.0850, 


*  See  Windisch,  Die  Bestimmung  des  Molecular-Gewichts  (Berlin, 
1892). 


1  68  PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 
and  from  this  the  molecular  weight, 
11.346  :  0.0859  —  132. 

Beckmann  gives  the  boiling-point  of  carbon  disul- 
phide,  CSa,  as  46°.  2  C.  ;  the  heat  of  evaporation  for  one 
gram  is,  according  to  Regnault,  84.82,  From  this  the 
proportional  factor  /"is  found  to  be 


It  is  a  function  of  the  boiling  temperature  ;  for  ex- 
ample, at  pressure, 

257       760        1841  mm,  of  mercury, 
or  boiling  temperature, 

16.2      46.2       76°.  2  C., 
the  molecular  increase  of  the  boiling-point  is 

18.9       24.0       30.4 

From  Beckmann's  observations,  1.4475  gr-  °f  phos- 
phorus in  54.65  gr.  of  carbon  disulphide  boils  at  a 
temperature  o°.486  higher  than  the  pure  solvent,  it  fol- 
lows now  from  the  approximate  formula  (15^)  that  the 
number  of  mols  dissolved  is 

0.486  54.65 

-^r-~r-  =0.0111, 
31.6      76 

and  the  molecular  weight  is  1.4475  "  0.0111  =  130,  so 
that  Pt  =  124  represents  a  molecule.  From  the  molec- 
ular weight  124,  the  number  of  mols  would  be  0.0117, 
and  the  increase  of  the  boiling-point  O°.5I4C.  follows. 


FREEZING  AND   BOILING  POINTS.  169 

From  an  observation  by  Arrhenius  a  water  solution 
of  sodium  chloride,  which  contains  0.273  gr.  Nad  in 
IGO  ccm.  H2O,  freezes  at  —  o°.i77C.  The  molecular 
lowering  of  the  freezing-point  of  water  solutions  is 

2-2?3'    g    =  18.66. 
ioo  X  79  X  87 

If  Nl  is  the  number  of  mols  of  salt  in  ioo  cc.,  we  find 
...       0.177  .  ioo 


while  0.273  gr.,  at  the  molecular  weight  58.5,  would 
give  0.0047  as  the  number  of  mols  dissolved.  The 
NaCl  must  therefore  be  looked  upon  as  almost  com- 

pletely dissociated,  in  this  water  solution,  into  Na  and 

Cl. 

A  very  much  stronger  solution  of  3.155  gr.  NaCl  in 
ioo  cc.  H2O  showed  a  freezing-point  of  —  i°.894.  It 
follows  that 

.  ioo 


is  the  number  of  mols  in  ioo  cc.,  while  the  molecular 
weight  58.5  would  lead  to  3.155  :  58.5  =  0.054.  In  this 
case  the  salt  is  not  completely  dissociated. 

If  the  degree  of  dissociation  (seepages  109  and  no) 
is  ?;,  we  find  that  instead  of  N'  molecules  we  have 


when  each  dissociating  molecule  separates  into  »'  mol- 
ecules. The  ratio  of  the  number  thus  found  to  be 
present,  to  the  number  that  would  be  present  if  there 


1 70  PRINCIPLES  Of  MATHEMATICAL   CHEMISTRY. 
were  no  dissociation  is  Nl :  N'  —j,  and  we  have 

y-  i  +  Ca7  — j>. 

In  the  above  example  «' —  2,/  =  0.102:0.054  = 
1.89,  rj  =  0.89. 

Notwithstanding  that  it  is  the  first  idea  in  our  pres- 
entation of  physical  chemistry,  to  derive  the  natural 
phenomena  from  the  general  subject  of  energetics, 
still  we  cannot  omit  the  general  theoretical  relations 
which  exist  between  the  single  phenomena  independent 
of  that  foundation  principle  of  all.  When  we  see  how 
the  single  facts  mutually  support  one  another,  the  value 
of  the  great  principle  underlying  them  all  will  be  recog- 
nized. Besides  this,  it  follows  from  the  fact  that  from 
one  the  others  may  be  derived,  that  the  phenomenon 
is  also  given  with  an  equal  degree  of  accuracy. 

Of  the  two  equations  (10)  and  (15),  that  concern  the 
evaporation  of  salt  solutions,  the  one  is  a  mathematical 
consequence  of  the  other.  If  we  represent  the  vapor 

pressure  /  of  a  solution,  as  a 
function  of  the  temperature  8, 
and  plot  it  as  a  curve  for  the 
pure  solvent,  we  obtain  Fig. 
n.  We  obtain  for  the  con- 
centration O,  as  well  as  for 
0  any  other  concentration  C1 ,  of 


0r      G  the  salt  another  curve  for  the 

FIG.  ii.  vapor  pressure :  /  is   here  a 

function  of  6  and  Ct.  Neglecting  the  higher  powers, 
Taylor's  series  gives 

f(e,  C)  =f(0.,  o}  +       (e  -  0.)  +  (\  c,. 


FREEZING   AND   BOILING   POINTS.  IJl 

In  order  that  the  solution  may  boil  at  the  same  pres- 
sure as  the  pure  solvent,  for  example,  at  atmospheric 
pressure,  it  is  necessary  that 


According  to  (10)  the  right  side  is  p9  .  £7,,  and  ac- 
cording to  (14),  page  91,  the  left  side  can  be  rear- 
ranged. We  find 


w 

i.e.,  equation  (15). 

This  consideration  for  the  process  of  evaporation 
could  be  carried  over  to  that  of  freezing,  if  the  func- 
tions of  the  temperature  in  the  figure  were  chemical 
intensities ;  then  the  above  development  would  lead 
also  to  (15)  as  applied  to  the  process  of  freezing. 

But  here  the  functions  are  vapor  pressures ;  a  solu- 
tion as  well  as  the  solvent  has  the  same  vapor  pressure 
when  boiling — the  atmospheric  pressure,  for  example  ; 
by  freezing  of  the  solution,  however,  the  partial  pres- 
sure of  the  vapor  does  not  necessarily  agree  with  that 
of  the  pure  solvent  by  freezing.  These  two  pressures 
are  related  in  another  manner :  they  belong  to  the 
curve  of  vapor  pressure  of  the  solid  phase  which  sepa- 
rates out  by  freezing. 

If  first  we  imagine  water,  and  then  ice,  at  the  tern- 


PRINCIPLES  OF  MATHEMATICAL    CHEMISTRY. 


perature   0,  transformed  into    vapor,    and    if   we   dis- 
tinguish the  two  curves  of  vapor  pressure  (Fig.  12)  by 


the  indices  w  and  z,  then,  by  use  of  equation  (14),  page 
91,  in  both  cases,  and  by  subtraction  of  the  heat 
—  (20  (that  is  necessary  to  change  I  mol  of  ice  into 
water),  we  find 


where//  is  the  pressure  of  the  vapor  at  #0°.  —  Q0  is 
used  in  order  to  bring  the  sign  the  same  as  it  was  above. 
Now  by  Taylor's  series,  neglecting  higher  powers, 


Here  /„  is  the  vapor  pressure  of  water,  and/  that  of 
ice  at  6°.  If  now  6°  is  the  freezing-point  of  the  solu- 
tion, then/  is  also  the  vapor  pressure  of  the  solution 
at  freezing.  The  equations  give 


FREEZING  AND   BOILING   POINTS.  I?3 

a  formula  which,  by  neglecting  the  small  difference 
between  //  and/,  corresponds  to  equation  (15^),  since 
from  io#  we  have  the  relation 


CHAPTER  V. 

OSMOTIC    PRESSURE. 

UP  to  the  present,  all  parts  of  the  systems  consid- 
ered have  been  restricted  by  the  condition,  that  equi- 
librium can  only  be  present  when  the  pressure  of  the 
whole  system  is  the  same.  It  is  possible,  however,  to 
realize  conditions  by  which  the  possibility  of  chemical 
action  is  present,  as  in  the  ones  already  discussed,  and 
by  which  even  after  entrance  of  equilibrium  the  pres- 
sure of  the  single  parts  is  different  and  remains  so. 
Gravity  causes,  for  example,  in  a  column  of  vapor  of 
considerable  height,  differences  in  pressure  of  that  kind. 
If  two  substances,  for  example  a  solution  and  the  pure 
solvent,  give  the  same  vapor,  at  the  same  temperature, 
under  different  vapor  pressures,  it  is  possible,  when  the 
cwo  substances  are  near  one  another  in  a  closed  space, 
for  equilibrium  to  be  present  only  after  their  surfaces 
have  stopped  at  different  height,  which  is  caused  by 
their  unequal  evaporation.  It  is  only  when  the  dif- 
ference in  height  of  the  evaporating  surfaces  is  equal 
to  their  difference  of  vapor  pressure  that  equilibrium 
can  be  reached. 

The  chemical  intensity  of  each  constituent  must 

174 


OSMOTIC  PRESSURE.  .     1/5 

also  in  such  a  case  be  the  same,  as  an  examination  ac- 
cording to  the  conclusions  reached  on  pages  115  and 
116  will  show. 

Experimentally  such  differences  of  pressure  can  be 
made  by  means  of  semi-permeable  cells,  as  Graham 
used,  and  as  Pfeffer  has  so  carefully  and  completely 
studied.  They  are  of  great  importance  ;  some  sub- 
stances can  go  through  them,  while  others  are  pre. 
vented.  In  other  words,  on  both  sides  of  such  a 
partition  the  chemical  intensities  of  certain  substances 
are  equal  when  equilibrium  has  been  established  ;  while 
the  chemical  intensities  of  other  substances  do  not 
equalize  at  all  and  yet  equilibrium  is  present.  If  on 
both  sides  of  a  semi-permeable  partition  there  is  a  sol- 
vent which  can  pass  through  it  readily,  and  if  we  replace 
that  on  one  side  by  a  dilute  solution  of  a  salt,  which  can- 
not pass  through  it,  then  the  intensity  of  the  solvent 
in  the  solution  must  be  just  as  great  as  in  the  pure 
state. 

We  will  derive  this  from  the  equation  of  energy  for 
the  solution  and  for  the  pure  solvent.  For  the  solu- 
tion we  have 

(1)  dE<  OdS  -  PdV+  IUM  +  II 
and  for  the  solvent 

(2)  dE9  <  6dS9  - 


P  and  P0  are  the  pressures,  V  and  V,  the  volumes, 
S  and  S0  the  entropies,  E  and  E^  the  energies  of  the 


i;      PRINCIPLES  OF  MATHEMATICAL    CHEMISTRY. 

two  bodies  on  the  two  sides  of  the  semi-permeable  par- 
tition, 6  their  common  temperature,  H0  the  chemical 
intensity  of  the  pure  solvent,  M0  its  mass,  II  and  M 
the  corresponding  values  for  the  solvent  in  which  the 
salt  is  dissolved,  and  TIl  and  Ml  the  intensity  and  mass 
of  this  substance.  If  now  the  two  phases  form  an 
isolated  system,  as  it  must  when  equilibrium  is  estab- 
lished, then  according  to  pages  115,  116,  and  117,  or 
also  page  77, 


Further,  none  of  the  volumes  can  change  without 
volume  energy  being  given  out  or  absorbed  ;  therefore 


dV=o,    dV.= 


and  the  nature  of   the   half-porous  partition   requires 
that 

(3.)  dM,  =  o. 

Hence  the  addition  of  the  above  energy  equations 
gives  us 


(4)  o(n 

and  since  dMt  can  be  positive  or  negative, 


The  intensity  is,  however,  a  function  of  the  concen- 


OSMOTIC  PRESSURE.  I// 

tration  and  pressure.  If  the  concentration  of  the  solu- 
tion is 

:c=^ 

where  n  is  the  number  of  mols  of  the  solvent,  »t  that  of 
the  salt.  If  P  is  the  pressure  on  the  side  of  the  solu- 
tion and  P0  that  on  the  side  of  the  pure  solvent,  then 
according  to  Taylor's  theorem  the  intensity  of  the- 
solvent  in  the  solution  is 


where  -ZT0  is  the  intensity  of  the  pure  solvent  to  which 
the  terms  with  the  index  o  refer.  Higher  powers 
are  neglected  since  the  solution  is  assumed  to  be  very 
dilute.  (If  this  latter  were  not  the  case  we  would  have 


Since  now  the  two  intensities  are  equal, 


it  follows  by  aid  of  equation  (26),  page  125,  and  equa- 
tion (21),  page  123,  from  equation  (6), 

(7)  ,(/>-/>„)  = 


C  —  I  =  —  nv  :  (n  +  ?0  =  —  Clt  where  Cl  is  the 
concentration  of  the  salt  in  the  solution.  Here  v  equals 
the  specific  volume,  m  the  molecular  weight  of  the  sol- 
vent, 6  the  absolute  temperature,  and  R^  the  gas  con- 


1  78   PRINCIPLES    OF  MATHEMATICAL    CHEMISTRY. 

stant  of  nearly  2  cals.    We  can  bring  in  still  the  volume 
of  a  mol  of  the  dissolved  substance,  vm,  so  that 

(8)  nl.vm—v.m.n, 

and  obtain 


The  difference  of  pressure  P  —  P0,  by  which  the  solu- 
tion is  stronger  than  the  solvent,  i.e.,  the  osmotic  pres- 
sure of  the  solution,  depends  upon  the  temperature  and 
the  volume,  which  the  dissolved  substance  occupies,  ac- 
cording to  the  same  laws,  as  if  the  latter  was  in  the  state 
of  a  perfect  gas.  This  law,  which  was  discovered  by 
van't  Hoff,  is  sustained  by  experiment. 

Pfeffer  observed  that  a  \%  sugar  solution,  by  use  of 
a  membrane  of  copper  ferrocyanide,  at  a  temperature 
of  I5°.C.,  gave  a  difference  of  pressure  against  water  of 
0.684  atmosphere.  Since  a  mol  of  raw  sugar,  C,aHaaOn, 
weighs  342  grams,  then 

0.684  .  1033.3  .  34,200  —  288.5  Ro 
is  to  be  substituted,  from  which  follows 

R0  =  83,800  g*cm.  :  °C.  =  1.94  cals.  :  °C. 
The  observation  that  at  o°  a  \<f>  sugar  solution  shows  a 
pressure  of  49.3  cm.  of  mercury  leads  to  R0  =  84,200  or 
1.96. 

A  mol  in  a  liter  would,  according  to  this,  at  o°, 
exert  a  pressure  of 

1.96  X  273 

-  -  -  -  -  =  22  atmospheres, 
1000  X  1033.3 

whether  the  substance  was  dissolved  or  not, 


OSMOTIC  PRESSURE. 


1/9 


From  the  laws  of  osmotic  pressure,  by  an  approxi- 
mate calculation,  we  find  the  laws  for  the  lowering  of 
the  vapor  pressure  of  a  dilute  solution.  A  vessel  A, 
closed  at  the  bottom  with  a  semi- 
permeable  partition  (Fig.  13), 
contains  a  solution,  and  its  lower 
part  is  placed  in  another  vessel 
B,  in  which  there  is  the  pure 
solvent.  The  whole  stands  in  a 
closed  space  C  that  is  filled  with 
the  vapor  of  the  solvent.  This 
vapor  shows  on  the  surface  of 
the  solution  the  pressure  /,  on 
the  surface  of  the  outer  vessel  = 
the  pressure /„.  If  the  solution,  FIG.  13. 

at  the  establishment  of  equilibrium,  is  h  cm.  higher 
than  the  solvent  from  which  it  is  separated  by  the  semi- 
permeable  film,  then,  according  to* the  well-known  law 
of  the  equilibrium  of  gaseous  bodies, 


(10) 


,.  ,7       dh 

—  dp  =  s  .  dh  =  — , 
v 


when  s  is  the  specific  gravity  and  v  the  specific  volume 
of  the  vapor.  If  this  vapor  follows  the  laws  of  Mark 
otte  and  Gay-Lussac,  and  if  m0  is  its  molecular  weight, 
then  we  have  further 


(II) 


(lib) 


180  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

On  the  other,  hand  the  osmotic  pressure  is  now 
equal  to  the  difference  of  vapor  pressures  increased  by 
the  pressure  of  a  column  of  the  liquid  of  the  height  h. 
If  the  difference  in  specific  weight  of  the  two  can  be 
neglected  and  the  specific  weight  of  the  solution,  which 
strictly  is  (nm  -f-  njn^)  :  n^m)  can  be  called  nm  :  nlvm^ 
we  obtain 

(12)  P-Pa= 


It  therefore  follows,  when  we  also  neglect  the  pres- 
sure of  the  column  of  vapor/—/,,  as  compared  to 
that  of  the  column  of  liquid,  from  equation  (9),  since 
m  =  m0, 

(13)  *•]•«.  =  x.e. 

We  obtain  by  use  of  equation  (u#) 


If  now/  and/0  are  but  slightly  different,  as  is  the 
case  with  dilute  solutions,  we  can  place 


and  we  find 

(15) 


p  n 

ite.,  equation  (io#),  page  164. 


OSMOTIC  PRESSURE.  l8l 

The  osmotic  pressure  has  obtained  a  very  promi- 
nent theoretical  place  in  physical  chemistry,  since  it  was 
used  by  van't  Hoff  in  order  to  make  clear  the  relation 
of  chemical  substances.  According  to  the  foregoing 
book,  the  chemical  intensity,  which  was  introduced  by 
Gibbs,  leads  to  a  more  general  method  of  treating 
chemical  phenomena.  The  theory  should  not  stop  at 
the  osmotic  pressure,  but  should  reach  further  back  still 
to  the  intensities.  Of  the  relation  of  osmotic  pres- 
sure to  intensity  we  can  predict  only  for  very  highly 
diluted  solutions.  From  equation  (21),  page  123,  it  fol- 
lows that  the  chemical  intensity  TIl  of  a  constituent  in 
highly  diluted  solutions,  is  related  to  the  osmotic  pres- 
sure of  the  solution,  as  against  that  of  trie  pure  solvent 
by  the  formula 


where  Al  is  a  function  of  pressure  and  temperature,  and 
./?,  is  the  specific  gas  constant  of  the  constituent. 


CHAPTER  VI. 

DIFFUSION. 

IN  the  previous  chapters  we  have  studied  the  equi- 
librium of  chemically  different  substances.  Before, 
however,  equilibrium  is  established  changes  of  energy 
take  place,  and  these  characteristic  quantities  of  the 
system  are  functions  of  the  time.  The  investigation 
of  these  changes,  during  a  certain  interval  of  time,  still 
remains  to  be  considered. 

In  mechanics  those  changes  of  energy  which  de 
pend  upon  the  time  are  classed  under  a  special  form 
of  energy,  i.e.,  kinetic  energy.  In  like  manner  the 
changes  in  a  system,  in  whose  different  phases  the  same 
constituent  has  a  different  intensity,  can  also  be  con- 
sidered under  this  heading  of  kinetic  energy.  In  fact 
we  always  find  movements  in  such  a  system,  and  some- 
times very  powerful  ones.  In  particular  we  must 
ascribe  to  kinetic  energy,  the  decrease  of  the  intrinsic 
energy  for  non-reversible  processes,  by  which  the  func- 
tion of  quantity  (entropy,  volume,  and  mass)  remains 
unchanged.  The  standpoint  of  every  molecular  hy- 
pothesis is  that  the  kinetic  energy  of  the  motion  is  the 
immediate  cause  of  the  chemical  reaction,  and  the 
phenomenon  of  motion,  shown  by  chemical  processes, 

182 


DIFFUSION.  183 

has  thus  been  the  principal  source  of  these  hypotheses. 

However,  this  has  not  proved  favorable  for  the 
mathematical  treatment  of  the  subject.  In  the  cases 
where  kinetic  energy  has  been  exerted  upon  the  sur- 
roundings it  has  proven  more  to  the  point  to  carry  the 
chemical  investigation  to  the  establishment  of  an 
"  overpressure  "  (iiberdruck) ;  or  in  other  words,  to  look 
upon  the  consequence  of  the  chemical  reaction  as  a 
pressure  energy,  the  calculation  of  which  into  kinetic 
energy  is  then  possible  by  means  of  the  principles  of 
dynamics.  The  osmotic  pressure  can  serve  as  a  good 
illustration  of  this  method. 

In  many  cases,  however,  no  kinetic  energy  is  given 
out  by  the  system,  and  in  others  so  little  as  to  be  un- 
noticeable,  as  compared  with  the  other  forms  of  energy, 
especially  heat ;  this  fact  has  led,  in  the  molecular 
hypothesis,  to  the  assumption  of  internal  friction. 

These  cases  of  motion  are  always  shown,  by  the  in- 
crease of  one  phase,  at  the  cost  of  another,  as  in  the 
change  of  place  of  a  phase  in  the  system* 

The  simplest  case  of  this  is  the  motion  of  a  soluble 
salt  in  its  solvent.  We  possess  no  other  mathematical 
grasp  of  this  process  than  that  offered  by  Pick's  law  of 
diffusion.  From  the  conception  of  chemical  intensity 
we  arrive  at  this  law,  which  was  deduced  experimen- 
tally, in  the  following  manner : 

We  imagine,  in  the  direction  of  length  of  a  cylin- 
drical tube  of  the  section  q,  a  small  separated  dis- 
tance dx.  It  encloses  a  small  cylindrical  volume  that 
contains  the  amount  of  the  substance  equal  to  pqdx, 
where  q  is  the  specific  gravity  of  the  constituent  of  the 


184  PRINCIPLES  OF  MATHEMATICAL   CHEMISTRY. 

solution  (or  the  substance).  This  amount  will  remaint 
immovable  as  long  as  the  chemical  intensity  77  of  the 
substance,  in  the  sections  immediately  touching  this 
section,  is  not  different  from  its  chemical  intensity. 
The  difference  of  intensity  in  the  distance  dx,  is  there- 
fore to  be  looked  upon  as  the  cause  of  the  motion  of 
the  constituent,  and  we  assume  that  the  velocity  u  of 
the  motion  in  the  direction  x  is  proportional  to  the  differ- 
ence of  intensity  in  this  direction  : 

CO  « =  -  h*j- 

The  proportional  factor  h  might  be  called  the  activ- 
ity *  of  the  constituent.  Its  unit  is  sec,  for  the  unit  of 
77  is  cma  :  sec2.  Since  by  page  123,  eq.  (21),  by  constant 
pressure  .and  temperature  6, 

(2)  LT=A  +  RBIC, 

where  A  is  a  constant,  C  the  concentration  of  the  con- 
stituent, and  R  its  specific  gas  constant.  We  obtain 
therefore 

If  n  mols  of  the  salt  are  dissolved  in  #0  mols  of  the 
solvent,  then,  when  C0  is  the  concentration  of  the  sol- 
vent, 

n  nn 


(4) 


'dC  nQ          3»          C0      c)n 


Beweglichkeit. 


DIFFUSION.  185 

If  we  use  instead  of  the  number  of  mols  n,  the 
specific  gravity,  A*,  of  the  salt,  i.e.,  the  number  of  grams 
in  cc.  (/.  /*  =  n  .  mt  where  n  is  referred  to  cc.  and  m  is 
the  molecular  weight),  it  follows  from  (3) 


Here  the  left  side,  /i«,  is  the  amount  of  the  constit- 
uent which  wanders  in  the  time  I,  through  the  section 
I  in  the  direction  x.  It  is  proportional  to  the  differ- 
ence of  density  of  the  constituent  in  this  direction  (or 
to  the  differential  quotients  of  the  density  taken  in 
the  opposite  direction). 

.  The  entrance  of  the  constituent  through  the  one, 
and  its  exit  through  the  other  section  of  the  volume 
considered,  of  the  length  x  increases,  as  is  self-evi- 
dent, the  amount  of  substance  in  each  cubic  centimeter 
between  the  two  sections  in  the  time  i,  by  the  amount 


The  hypothesis  (i)  that  has  thus  led  us  to  the  law 
of  diffusion  is  in  complete  accord  with  the  formulae  for 
the  diffusion  of  other  forms  of  energy.  If  instead  of 
the  chemical  intensity  we  substitute  the  temperature, 
the  pressure,  or  the  electrical  potential,  we  obtain  the 
well-known  and  proven  laws  for  the  forms  of  energy. 

The  diffusion  constant  k  of  eqs.  (5)  and  (6)  has  the 
value 


(7)  k  =  h.Rd.Cl 


0> 


1  86  PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 
and  for  highly  diluted  solutions 
k  =  h.  RB. 


Its  unit  is  found  from  that  as  from  other  formulae  to 
be  cm.a  :  sec.  Experience  shows  that  /i,  for  different 
substances,  is  different,  and  also  has  a  value  depending 
upon  the  temperature,  and  Nernst  has  succeeded  in 
bringing  the  value  of  h  for  electrolytes,  in  a  form  con- 
sistent with  the  theory.  When  we  remove  from  his 
calculations*  the  molecular  starting-point,  it  is  essen- 
tially as  follows  : 

We  assume,  with  Arrhenius,  that  each  electrolyte 
is  formed  of  two  constituents,  equally  mixed,  which 
change  their  positions  by  electrolysis  ;  we  therefore 
have  two  intensities,  T1A  and  TIK,  instead  of  77,  so  that 
in  the  energy  equation,  instead  of  UdMy  we  have  the 
sum  TlAdMA  -\-  HKdMK.  Since,  however,  the  changes 
of  mass  dM,  dMA  ,  and  dMK  of  the  electrolytes  depend 
upon  the  electrochemical  equivalents  a,  a1  ,  a,  of  these 
substances,  we  have 

(8)  Ua  =  UAa,  +  77^2. 

Nernst's  theory  considers  now  the  diffusion  of 
electrolytes,  as  the  diffusion  of  these  two  constituents. 
If  /zt  and  //,,  are  the  "activities"  (Beweglichkeiten)  of 
the  ions,  then  the  velocity  of  their  motions,  as  in  the 
case  of  ordinary  diffusion,  will  be 

(9)  ***  *- 

*  Nernst,  Theoretical  Chemistry,  trans,  by  Palmer,  Ger.  ed.  p. 
309  and  fol. 


DIFFUSION. 


If  these  equations  are   multiplied  by  ajt^  and  a^ 
respectively,  and  added,  it  follows,  by  aid  of  (&'),  that 


(10) 


The  comparison  of  (i)  with  (/£)  gives  the  diffusion 
constant  for  high  dilution  as 

^.*- 

(») 


Upon  the  "  activities "  /^  and  ^3 ,  however,  the 
electrolytic  phenomena  depend. 

In  electrolysis  the  velocities  ul  and  u^  of  the  ions 
are  no  longer  equal,  as  with  the  ordinary  diffusion  of 
electrolytes.  It  is  not  in  consequence  of  a  difference 
in  concentration,  but  in  consequence  of  a  difference  of 
the  electrical  potential  P,  that  now  the  chemical  in- 
tensity of  each  ion  varies  from  place  to  place,  and 
causes  motion.  If  the  ordinary  diffusion  does  not  take 
place  at  the  same  time  as  that  by  electrolysis,  then  the 
intensities  differ  only  in  consequence  of  the  potential 
Pj  and  are  functions  of  this  potential  alone. 

Equation  (i)  gives,  therefore,  of  ion  velocities 


(12) 


1  88  PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY, 

In  order  now  to  find  the  differential  quotients  of  the 
intensity,  according  to  the  electrical  potential,  we  follow 
the  same  method  as  we  used  on  page  124  for  the  cor- 
responding case.  We  start  from  the  energy  equation 
(12),  page  107, 

(13)         dE=  BdS  -  Pde  +  TiAdMA  +  HKdMK  , 

in  which  dE  is  the  energy  passing  through  the  section 
in  the  unit  of  time,  de  the  amount  of  electricity,  dMA 
and  dMK  the  amounts  of  the  anion  and  kathion  going 
through,  dS  the  entropy,  and  6  the  temperature.  We 
write 


(14)  d(E  +  Pe)  =  MS  +  edP+  TlAdMA 

and  conclude  from  the  property  of  a  complete  differen- 
tial that 


Now  the  amount  of  electricity  is,  according  to  page 
loi,  bound  to  the  ions,  and  just  as  much  on  the  anion 
as  on  the  kathion,  and  on  each  equivalent  of  the  elec- 
trolyte the  amount  xe.  ==  — ,  where  x  =  -  —  1 .036 .  IO~U, 

eo  G» 

and  e  is  measured  in  absolute  units.     We  have 

+ 


DIFFUSION.  189. 

and  from  equation  (12)  and  (15) 

i      dP  i      dP 


, 

(17)     u  ,  =  —  h. 
1 


,       - 


On  the  other  hand,  it  follows  from  equation  (i8V 
page 


where  /ix  and  ^  are  the  molecular  conductivities  of  the 
ions.     The  comparison  gives 


.  . 

(19)  —±-  =  2XK,      — - 

xat  xan 


or  according  to  page  108,  by  using  the  conductivities 
observed  by  Kohlrausch  and  the  degree  of  dissocia- 
tion 77, 

h.  looo     h»  ..  1000 

(iQ^) — — =2yW1    ..#  .  0"Hg. ,    =  2^3     .X  .  O"Hg  . . 

xal  rf       xa^  rj 

By  using  these  results  we  obtain 

h                                             ,.           6   1000 
-  =  2^* .  1.036 .  io~4 .  1.063  .  io~6 . ; 


—^  =  2/<*  .  1.  10.  I0~6-,  —  ^  =  2/*0*  .    1  .10.  IO"6- 

17  ^a 


UNIVERS5TTV 


PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

If  the  electrolyte  is  completely  dissociated  into  ions, 
i.e.,  rj  •=  i,  it  follows 


(20)  A   =  2;*,*  .  1.  10  .   I0-8,     A   =  2p*  .  1.  10  . 
xa^  xa^ 

If  we  write  now  by  page  27 


IO-. 


then   for  univalent  substances,  by  which  the  mol  and 
equivalent  are  the  same, 

n 

aRB  =  2.270  —  .  io10; 
and  it  follows  from  (n) 

(22)      k=        »'7<a*»   .2.1.10.   IO-«.  1.036.  I0~4.  2.270.   10".—. 
/*i*  +  /*.*  273 

Since  the  value  of  /**  for  18°  C.  has  been  observed,  it 
is  better  to  use  6  =  273  -f  $  =  291  +  ($  —  18);  hence 


2'27°  ' 


*  =  5.5,5  -  .  [i  +  0.0034(*  -  18)]. 

The  cm.8  :  sec.  is  transformed  into  cm.a  :  days  by  multi- 
plication by  8.64  .  io4. 

23fr       *  =  0.477  .  io<  .  .  [i  +  o.oo34(^  -  18)] 


If  we  choose,  for  example,  with  Nernst  the  values 
4.2   and  6.3  (i  8°)  for  Na  (/^lo6)  and  Cl  (^*iOe),  it 


DIFFUSION.  IQI 

follows  for  NaCl  that  k  =  1.20,  while  Scheffer's  obser- 
vation gave  i,  1 1. 

The  increasing  of  the  molecular  conductivity  by 
about  2  per  cent  for  each  degree  of  temperature  leads 
to  an  increase  of  k  of  0.023  X  its  value  for  each  degree, 
which  agrees  very  well  with  the  observation. 


CHAPTER  VII. 

THE  VELOCITY  OF  A   CHEMICAL  REACTION. 

THE  assumption  that  the  velocity  of  diffusion,  u,  is 
proportional  to  the  difference  of  chemical  intensity, 


allows  us  deeper  insight  into  the  subject.  We  can 
assume  that,  in  analogy  to  the  behavior  of  the  intensi- 
ties of  other  forms  of  energy,  each  chemical  intensity 
strives  to  expand.  If  there  are,  in  the  two  sections 
that  bound,  at  the  distance  x,  a  cylindrical  volume, 
two  chemical  intensities  I11  and  77,  ,  then,  according  to 
this  view,  the  diffusion  in  this  volume  is  caused  by  the 
two  different  forces,  each  striving  to  expand  its  own 
chemical  state  ;  and  the  stronger  must  succeed.  By 
this  we  introduce  no  new  method  of  consideration  ;  but 
only  apply  to  chemical  phenomena  the  ordinary  repre- 
sentation of  the  process  of  diffusion. 

In  chemical  processes  we  find  but  one  objection  to 
this,  that  is,  that  though  the  expansion  may  be  one  of 
position,  as  in  diffusion,  it  is  not  compulsory  that  it 

192 


VELOCITY  OF  A    CHEMICAL  REACTION. 

should  be  so.  In  the  case  of  the  inversion  of  sugar, 
/or  example,  the  chemical  reaction  takes  place  regularly 
and  uniformly  in  all  parts,  and  as  the  time  increases 
the  number  of  mols  inverted  also  increases.  If  A7"  is 
the  number  of  mols  that  are  present  at  the  chemical 
intensity  II,  then 

dN 

(i)  =  kn, 


where  k  is  a  constant  proportional  factor.  That  is,  the 
increase  in  the  number  of  mols  which  enter  into  reaction 
in  the  time  dt  is  proportional  to  the  intensity.  Accord- 
ing to  pages  133  and  134  the  intensity  of  a  reaction  is 
composed  of  the  intensities  of  its  reacting  constitu- 
ents, and  if  we  measure  the  progress  of  the  reaction  by 
the  change,  in  unit  time,  of  the  number  of  mols  A"0  of 
any  of  the  changing  constituents,  as,  for  example,  one 
which  decreases  by  the  reaction,  it  follows  for  constant 
pressure  and  temperature,  by  aid  of  eq.  (21),  page  125, 

(2)    -  -^  =  W/(C<'  •  •  •  C)  +  *„• 

Here  #0  is  a  quantity  dependent  only  upon  the 
pressure  and  temperature.  Cis  the  concentration,  v  the 
exchange  numbers  of  the  n  reacting  constituents,  6  the 
absolute  temperature,  R0  the  general  gas  constant, 
k0  a  constant  whose  value  depends,  among  other  things, 
upon  the  choice  of  the  constituents,  according  to  the 
molecular  weight,  N9  ,  by  which  the  progress  of  the  re- 
action is  measured. 


194  PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 

After  entrance  of  equilibrium  the  concentration 
product  has  become  /£0,  a  value  depending  upon  pres- 
sure and  temperature.  It  is 

p  =  J^,AX'..+  *A; 

and,  by  subtracting  this  from  the  former  we  eliminate 
#,  and  obtain 

(  *\  -       °  —  k  R  61  X  C  v*  C  *'»  Cv* 

V3/  ~dt~   ~  K  •  ••«•*•• 

If  now  we  designate  those  constituents  which  have 
positive  exchange  numbers  (i.e.,  those  that  increase 
during  the  reaction)  by  the  indices  «,$,£...,  and 
the  decreasing  ones  with  a,  /?,  y  .  .  .,  it  follows 


(4)  -         =  kJi. 


We  will  only  follow  the  subject  further  under  the 
condition  that  the  two  concentration  products, 

c  —  CVaC  vt>  n  — 

C  -    L'a      t-i         •     .     .,         '/  — 

differ  but  slightly  from  one  another,  i.e.,  that  the  reac- 
tion goes  very  slowly,  a  condition  that  must  be  ful- 
filled anyway  on  account  of  the  constant  temperature. 
In  this  case 


_£ 


VELOCITY   OF  A    CHEMICAL   REACTION.         195 

and  since  -  —  changes  but  slightly  during  the  reac- 

tion, and  by  neglecting  the  small  quantities,  of  a  higher 
order  than  c  —  yr,  we  can  write 


._.. 

-  dt    '-  c(C 


where  M0  and  M0   —  K0M0  are  constants. 

Finally,  we  introduce  the  same  term  as  we  did  on 
page  142.  We  will  measure  the  amounts  of  the  con- 
stituents that  were  present  at  first,  and  which  increase, 
in  equivalents  and  call  them  A,  B...,  their  exchange 
numbers  are  a,  b  ;  the  equivalents  present  at  first  of 
the  decreasing  substances  are  then  A,  B  .  .  .  ,  and  their 
exchange  numbers  a,  ft,  y  .  .  .•  and,  finally,  we  will  des- 
ignate by  x  the  proportional  number  of  equivalents, 
n0,  which  have  reacted  during  the  time  t  seconds. 
Then 


(6)      =- 


where  M  and  M   are  new   constants,  and  M  =  MK, 
when  K  depends  only  upon  pressure  and  temperature, 
and  condition  the  entrance  of  equilibrium  (page  143). 
If  x  equivalents  are  transformed,  then 


The  law  according  to  which  a  slow  reaction  takes 
place  is,  under  all  circumstances,  expressed  by  the  inte- 


196  PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

gral  of  a  rational  function,  i.e.,  by  logarithms  and  ra- 
tional functions.  The  law  of  Guldberg  and  Waage, 
page  143,  is  a  special  form  of  eq.  (6).  At  dx  :  dt  —  o 
the  kinetic  process  becomes  a  static  one. 

If  a  reaction  takes  place  in  one  direction  only,  so 
that  equilibrium  is  reached  only  after  the  constituents, 
which  decrease  by  the  reaction,  are  all  used  up,  so  that 
^=00,  i.e.,  M  negligibly  small  as  compared  to  M, 
then  the  equation  is 

(;)  =  -  M(A  +  *Y(P  +  *?  -r  #  , 


or 

~ 


(7b) 


The  first  is  for  a  finite  value  of  M,  and  the  latter  for 
very  small  M:  in  the  first  case  M  is  so  large  that  the 
change  in  the  value  of  x  is  without  influence  upon  the 
second  term  ;  this  therefore  retains  a  constant  value  N. 
The  oldest  confirmation  of  the  theory  are  the  re- 
sults of  the  experiments  of  Wilhelmy,  1850,  on  the 
inversion  of  raw  sugar.  In  a  water  solution  this  falls  into 
Dextrose  and  Levulose,  according  to  the  formula 

CnH21On  +  H,0  =  C.H..O.+  C.H.A. 

Since  a  state  of  equilibrium  is  only  reached  here 
after  complete  disappearance  of  the  sugar,  eq.  (7$) 
must  be  used.  Neglecting  the  concentration  of  the 
water,  we  find 


VELOCITY  OF  A    CHEMICAL  REACTION.        1  97 

(8)  d£=U.(A-x), 

where  A  is  the  initial  amount  of  sugar  in  the  solution, 
and  x  the  amount  inverted  in  the  time  /. 

It  is  not  necessary  here  to  measure  these  amounts 
by  equivalents,  for  there  is  but  one  substance  present, 
and  we  can  measure  it  as  well  by  grains  or  any  other 
unit.  By  integration  we  find 


In  order  to  measure  the  amount  of  sugar  inverted 
Wilhelmy  used  the  polariscope.  Since  the  angle 
through  which  the  analyzer  is  turned  is  proportional 
to  the  amount  of  sugar,  these  angles  can  be  used  in- 
stead of  the  weight  or  equivalents.  He  found,  for 
example,  after  lapse  of 

o  15  60  105  120       oo  minutes 

the  angle  was 

46°.  75  43°-75  35°-75   28°.25   26°.o   -i8°.7O 

We  have  then 

A  -  x  -  65.45       62.45          54-45          46.95         44-70  o 

10lo£  —  =       O  O.O2O38        O.O799I        O.I4427        0.16560  oo 

A  "™~  x 

->oi0g  —  A——  0.00136      0.00133      0.00138      0.00138 

i  A  —  x 

from  which  the  truth  of  the  law  can  be  seen. 

Ostwald  has  measured  the  accelerating  action  of  an 
acid  (catalytic  action)  upon  the  decomposition  of  an 


PRINCIPLES   OF   MATHEMATICAL    CHEMISTRV. 

ester  in  a  water  solution,  by  titration.  He  adds,  for 
example,  to  i  c.c.  methyl  acetate  10  c.c.  HC1  (i  mol  in 
liter),  dilutes  with  water  to  15  c.c.  and  titrates  a  sample 
after 

o       14       34       199       539        oo       minutes 
with  dilute  barium  hydroxide  solution.     At  first 
13.33  c.c. 

of  Ba(OH)a  were  needed  to  neutralize  the  free  acid, 
but  during  the  catalysis  the  amount  of  acetic  acid 
formed,  made  it  necessary  to  add 

0.92         2.14         8.82         13.09         14.  1  1  c.c. 

more  than  at  first.  The  amount  of  ester  originally 
present  is  represented  by  14.11,  and  we  find  of  A  —  x 

14.11  13.19         n-97        5'29         1.02          o 

and  by  use  of  logarithms 
log    -  =    0.0293       0.0714       0.4261       1.1409        o 

A  ^~  X 


-  log  — 


=   0.00209    0.00210    0.00214    0.00212 


so  that  the  value  which  should  be  constant,  according 
to  the  theory,  is  so  within  the  experimental  error. 

The  saponification  of  an  ester  is  an  example  of  a 
reaction,  by  which  two  substances  change  their  concen- 
tration, and  leads  to  a  complete  reaction,  i.e.,  one  to 


VELOCITY  OF  A    CHEMICAL  REACTION.        199 
which  equation  (7$)  applies.     According  to  the  formula 
C9H,COOCH8  +  NaOH  =  CH3COONa  +  C2H6OH. 


Ethyl  acetate  C^ic          =          *™  Ethyl  alcohol 


the  ester  changes  place  in  water  solution  with  the  base. 
The  amount,  changed,  in  mols,  for  the  time  t  can  be  de- 
termined by  titration  of  the  base,  and  if  A  and  B  are 
the  number  of  mols  of  ester  and  base  originally  pres- 
ent, then 


by  transforming  it  into 

**  /__£_ 


B  -  A  \A  - 

it  is  apparent  that  the  integral  of  the  differential  equa- 
tion, which  has  /  =  o  when  x  =  o,  is 

1      !B  ~  *  -  -  M/ 
B-A  A-  x  B~ 

If  at  first  the  base  and  ester  are  present  in  equiva- 
lent amounts,  i.e.,  A  =  B,  then  the  differential  equation 
is  simplified  to 

dx 
di 
and  its  integral  is 


Nernst  gives  the  following  series  of  observations  to 
illustrate  the  equation  which  holds  for  A^B. 


2OO  PRINCIPLES  OF  MATHEMATICAL    CHEMISTRY. 

The  reaction  as  given  above,  at  10°,  after 
/  =     o  4.89         10.37         28.18         oo  minutes, 

when  100  cc.  was  titrated  with  -  ^  normal  acid,  gave 

7*=  61.95        50.59        42.40         29.35  14.92  cc. 

The  division  of  these  numbers  by  23.26  give  the 
number  of  mols  in  a  liter,  of  the  base,  still  present,  i.e. 
B  —  x.  The  value  for  t—  oo  gives  B  —  A.  We  obtain, 
therefore,  by  logarithms 


=   I  _  . 


_  . 

14.92*  61.95    T  —  14.92 

which  gives 

0.0236        0.0238         0.0233 

As  an  example  of  the  general  equation  (6)  the  for- 
mation of  an  ester  can  serve  very  well.  The  mixture 
of  i  mol  of  alcohol  with  I  mol  of  acetic  acid  causes  the 
formation  of  ethyl  acetate  according  to  the  formula 

CH8COOH  +  CaH6OH  =  CH3COOCaH6  +  H9O. 

The  number  of  mols,  x,  used  during  the  time  / 
must,  where  none  of  the  final  product  was  present  at 
first,  satisfy  the  equation 


- 
at 

the  integral  of  which  (for  /  =  o,  x  =  o)  is 

log  £_£.*, 


VELOCITY  OP  A    CHEMICAL  REACTION.        2OI 
where  a  and  b  are  the  roots  of  the  equation 

•-          M  M 

2M-  M*  ^U-M" 

Since  in  the  case  considered  equilibrium  is  attained 

after  decomposition  of  2/3  of  a  mol,  i.e.,  —. -  =  o  for 

at 

x  =  2/3,  it  follows 

M=4>     a  =  2>     *  =  2/3'> 

4(M  -  M)  =  -  log  ?—^L. 
3  *         2  -  3* 

Guldberg  and  Waage  have  proven  this  formula  and 
found  only  a  fair  agreement  with  the  theory,  which 
would  have  been  much  better  if  x,  for  the  completed 
reaction,  had  not  been  taken  as  exactly  2/3,  but  had 
been  determined  each  time  by  observation. 

Further  examples  can  be  found  in  the  works  of 
Ostwald  and  Nernst,  which  have  been  mentioned  pre- 
viously. 


PART    IV. 

THE    DEGREES    OF    FREEDOM    OF 
CHEMICAL   PHENOMENA. 


CHAPTER  L 

THE   RULE  OF   PHASES. 

BY  the  equation 
dE  5  BdS  -  PdV  +  n^dM,  +  II9dMt  +  .. 

which  holds  for  any  homogeneous  body,  each  possible 
change  is  made  dependent  on  those  changes  which  are 
necessary  to  its  definition.  The  state  of  a  body  is  de- 
termined when,  besides  the  n  amounts  of  substance  M1  y 
M.2  .  .  .  Mn  (which  are  independent  of  one  another),  the 
temperature  6  and  pressure  P  are  known,  provided  that 
besides  changes  of  substance  only  changes  in  heat  and 
volume  take  place,  and  especially  that  those  in  elec- 
tricity, gravity,  and  surface  energy  are  excluded.  The 
n  +  2  variables  can  occur  in  the  same  manner  in  the 
energy  equation  by  writing  it  as  follows  : 

d(E-OS  +  PV)  <SdO-  VdP  +  II.dM, 


202 


THE  RULE   OF  PHASES.  20$ 

Here  all  the  terms  on  the  left,  S,  V,  Hlt  JT2,  .  .  .  Uni 
are  functions  of  0,  P,  M,  M^...  Mn. 

If  we  ask  what  possible  changes  a  certain  amount, 
say  I  gram,  of  the  homogeneous  body  is  capable  of  un- 
dergoing, the  answer  is  n  -f  I,  since  by  condition 
Ml  -f-  J/a  +  •  •  •  +  Mn  —  I  gr.  In  this  way  the  possible 
states  of  a  gram  of  water  has  a  freedom  of  two  dimen- 
sions; only  pressure  and  temperature  can  vary.  On 
the  other  hand,  i  gram  of  a  mixture  of  oxygen  and 
hydrogen  can  vary  in  three  ways  —  in  temperature, 
pressure,  and  composition. 

The  case  is  slightly  different  when  two  phases, 
which  contain  the  same  n  substances  independently 
variable,  form  a  system  between  which  exchange  of 
substance  is  possible.  We  have  from  page  115 


dE'  <  tfdS  -  P'dV  +  77,  ' 

+  II9 

dE"  <  &'dS'  -  P'dV"  + 


that  is,  2(n  +  2)  variables,  viz.,  6',  P't  Mt't  Mf  .  .  .  MJy 
and  0",  P\  M^,  J//'.  .  .  Mn".  We  know,  however,  that 
between  them,  in  the  case  of  equilibrium,  there  are 
n  +  2.  conditions. 

ff  =  0",  P  '  =  P",  77/  =  IT/',  77/  =  I7a/r,  ...nn'  =  IIn". 

In  case  of  equilibrium  the  mass  of  each  phase  is  a 
fraction  of  the  total  mass,  dependent  on  the  tempera- 
ture, pressure,  and  composition.  This  may  be  looked 
upon  here  as  an  experimental  law,  which,  however,  we 


204  PRINCIPLES   OF   MATHEMATICAL   CHEMISTRY. 

will  prove  from  our  theoretical  standpoint  in  the  next 
chapter.  If  the  total  mass  amounts  to  i  gram,  then 
the  expressions  Mv'  +  Mt'  +  .  .  .  +  Mn'  and  M,"  .  -f 
M"  +  •  •  •  +  MJ'  must  be  certain  functions  of  the 
other  variables.  As  independent  variables  we  can  take 
the  above  2(72  +  2)  quantities  only  so  long  as  all,  even 
those  which  cancel  at  the  equilibrium  of  the  two  phases, 
are  considered.  When  the  still  possible  changes  are 
not  to  disturb  the  equilibrium,  there  would  remain  only 


variables.  For  example,  the  states  of  a  gram  of  fluid,  as 
well  as  one  of  gaseous,  water  can  be  changed  each  in  two 
ways.  If  the  two  phases  are  brought  together,  then  to 
the  four  variables  a  fifth  is  added,  viz.,  the  exchange  of 
substance  from  one  state  to  the  other.  These  two  grams 
of  water  do  not,  however,  in  general  remain  in  equilib- 
rium. If  after  entrance  of  a  state  of  equilibrium  it  is  to 
be  retained  during  changes,  then  the  three  conditions 
necessary  are  equality  of  pressure,  temperature,  and 
chemical  intensity  in  the  two  phases  ;  and  a  fourth  is 
given  from  the  fact  that  the  pressure  is  a  function  of 
the  temperature,  by  the  law  of  vapor  tension,  and 
consequently  the  composition  of  the  two  phases  is 
determined.  The  mixture  has  therefore  but  one  degree 
of  freedom  ;  if  one  of  the  variables  is  known  as  the 
temperature,  then  we  know  all  the  others. 

We  can,  from  the  simple  case  given  as  an  example, 
also  show  it  geometrically,  when  in  addition  to  pres- 
sure and  temperature  we  use  the  chemical  intensity  of 


THE  RULE   OF  PHASES.  2O$ 

I  gram  of  water  as  a  coordinate.  For  one  gram  of  fluid 
water  and  one  in  the  gaseous  state  we  obtain  two  differ- 
ent surfaces  in  the  pressure-temperature  plane.  The  two 
intersect  in  a  curve,  i.e.,  where  the  pressure  and  tem- 
perature is  given,  by  which  liquid  and  gaseous  water 
can  exist  together,  and  by  which  the  two  phases  can 
exchange ;  the  states  in  which  this  is  possible  form 
a  system  of  one  dimension.  For  the  general  case  of 
a  freedom  of  more  than  two  dimensions  we  can  do 
nothing  geometrically. 

For  r  phases,  finally,  by  which  there  are  r(n  -\-  2) 
variables,  since  in  each  two  phases  the  temperatures, 
pressures,  and  intensities  must  be  equal,  we  find  (r  —  i) 
(n  -j-  2)  conditions,  and  by  the  fact  that  the  mass  of 
each  phase  is  a  certain  fraction  of  the  total  mass,  de- 
pending upon  the  temperature,  pressure,  etc.,  we  find 
still  r  conditions.  We  have  therefore  remaining 

r(n  +  2)  —  (r  —  i)(n  +  2)  —  r=n  -\-2-r 

independently  varying  quantities.  Therefore,  when  n 
substances  are  present  in  r  different  phases,  and  en- 
ter into  exchange  of  substance,  the  equilibrium  remains 
by  n-\-2  —r  different  kinds  of  possible  changes,  or 
the  equilibrium  of  substances  in  r  phases  is  a  state  of 
n  -\-  2  —  r  dimensions. 

This  law,  discovered  by  Gibbs,  is  not  changed  when 
in  a  single  phase  less  than  n  substances  are  present. 
We  can  in  all  phases  imagine  the  maximum  of  n  sub- 
stances and  assume  the  terms  dM  partly  equal  to  zero 
(without  retaining  them). 


206  PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 

In  the  pressure-temperature  plane  we  can  also  rep- 
resent in  general  the  proportions,  as  far  as  concerns 
changes  of  I  and  2  dimensions.  If  only  one  substance 
is  present  it  must  be  in  2  phases,  2  substances  must  be 
in  3  phases,  and  n  substances  must  be  present  in 
n  +  I  phases,  if  the  equilibrium  is  to  be  changed  uni- 
dimensionally,  and  when  to  each  pressure  the  tempera- 
ture is  to  be  determined  or  a  curve  in  the  pressure- 
temperature  plane  is  to  represent  the  possible  states. 
If  we  have  represented  in  this  plane  the  curves  of  uni- 
dimensional  (so-called  complete)  states  of  equilibrium, 
then  between  them  lie  those  states  by  which  pressure 
and  temperature  can  be  chosen  independently  of  one 
another,  i.e.,  by  presence  of  n  substances,  in  at  most  n 
phases,  in  a  state  of  exchange.  How  the  phases  pres- 
ent are  to  be  divided  among  the  n  single  fields  of  the 
plane  is  given  in  part  by  the  condition  that  in  each 
division  line  of  a  field  n  -j-  I  phases,  and  in  each  in- 
tersecting point  of  the  boundary  line  n  -f-  2  phases  must 
come  together,  the  latter  because  by  the  point  each 
possibility  of  change  is  excluded,  or  occurs  in  a  state  of 
zero  dimensions. 

One  glance  at  Fig.  5,  page  87,  for  the  possible 
states  of  a  single  substance  will  make  all  clear. 

The  cases  in  which  2  substances  form  a  number  of 
phases  have  been  investigated  by  Roozeboom.  Sul- 
phurous acid  and  water  form  the  following  phases : 
7,  ice  ;  H,  solid  hydrate,  SO2  +  ;H2O  ;  S,  liquid  acid 
with  small  amounts  of  water;  //,  water  solution  of  sul- 
phurous acid  ;  and  finally  a  gas  G,  composed  of  water 
vapor  and  gaseous  sulphurous  acid.  Two  liquid  phases 


THE  RULE   OF  PHASES, 


must  be  different,  because  water  and  liquid  SO2  exist 
together  somewhat  as  ether  and  water,  and  like  these 
substances  dissolve  one  another  slightly.  The  5  phases 
cannot  exist  together  in  equilibrium  ;  each  four  exist 
in  the  single  points,  of  which  Roozeboom  has  investi- 
gated 2,  each  3  in  curves  that  go  from  the  points,  and 
each  2  in  the  fields  between.  So  much  as  has  been 
studied  is  shown  in  Fig.  14. 

The  fourfold  point  B,  at  which  the  phases  7,  H,  L, 
and  G  coexist,  has  the  coordinates  $  =  —  2°.6,  p  = 
21. T  cm.;  and  the  fourfold  point  C,  by  which  //,  L,  S, 


FIG.  14. 

and  G  coexist  has  $  =  I2°.i ,/  =  177  cm.  AB  gives 
the  relation  between  pressure  and  temperature  that 
must  be  held  in  order  that  a  water  solution  of  sulphur- 
ous acid,  in  which  there  is  ice,  is  in  equilibrium  with  its 
vapor. 

The  strength  of  acid  in  the  water  becomes  smaller 
as  the  pressure  becomes  less,  that  in  the  vapor  becomes 
larger,  so  that  the  curve  A  nears  the  melting-point  of 
ice.  The  curve  BC  is  the  curve  of  vapor  tension  of  a 


2O8  PRINCIPLES  OF  MATHEMATICAL    CHEMISTRY. 

water  solution  of  sulphurous  acid  which  is  in  contact 
with  solid  hydrate,  i.e.,  the  curve  of  vapor  tension  of 
the  concentrated  solution ;  the  dotted  continuation 
BB'  is  the  non-reversible  change  of  state,  the  so-called 
overcooled  state. 

In  this  way  the  figure  can  be  understood  by  the  in- 
itials of  the  different  states.  How  HL,  HS,  and  HG  are 
transformed  into  one  another,  was  not  investigated,  it  is 
plainly  a  state  of  the  system  which  is  not  determined 
by  pressure  and  temperature,  and  so  the  mixing  propor- 
tions of  H2O  and  SO2  must  be  taken  as  a  third  variable. 
We  would  find  there  states  in  which,  by  a  given 
pressure  temperature,  two  different  pairs  of  phases 
could  exist  ;  as  well  as  those  in  which  pairs  of  phases 
could  not  exist,  they  consist  of  the  single  substances 
separated  from  one  another. 


CHAPTER   II. 

THE   EQUILIBRIUM   OF  PHASES. 

ON  page  203  we  used  a  law  as  an  experimental  one  ; 
we  will  now  derive  it  theoretically  in  the  way  that  its 
discoverer,  Gibbs,  did.  The  possible  changes  which  a 
part  M'  of  a  single  phase  in  a  system  can  undergo  must 
satisfy  the  equation 

(i)         dE'  5  6dS'  -  PdVf  +  IIJM'  +  njMj 


where 

M.'+M,'+  .  .  .  +  MH'  = 

The  equation  can  be  transformed  into 

(2)  -  d\&  -  es' 


i'dn.+Mldllt  +  ...  +Mn'dMnt 
always  under  the  condition  that 


209 


210  PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 

Since  now  by  transposition  from  one  phase  to  an- 
other the  differentials  on  the  right  are  all  o,  the  func 
tion 

F=E'-esf  +  PV-HM  -  zr,j//  4-  ...  nnM,; 

(which  is  called  the  thermodynamical  potential  or  the 
free  energy  at  constant  pressure,  temperature,  and 
chemical  intensity)  has  the  same  value  as 


as  has  each  corresponding  function  of  this  form  to  every 
other  phase,  provided  that  the  transformation  is  re- 
versible and  the  condition  holds  that 

M'  =  M"  =  .  .  . 

On  this  ground  we  can  answer  the  question,  as  to  how 
reversible  changes  of  pressure,  temperature,  and  inten- 
sity must  be  composed  in  order  that  the  possibility  of 
coexistence  of  each  phase  remains.  The  functions 
F't  F"  .  .  .  must  have  the  same  value,  and  when  this 
common  function  is  represented  by  F  the  equation 
must  hold  that 


(4)  dF=dEf  -  6dS'  +  PdV  -- 

-  ...  nndMnf  -  S'dO  -\-VdP- 


that  is,  by  (i), 

(5)        dF  =  -  S'dO  + 

-  M  ' 
—  .  .  ,        ivj. 


THE  EQUILIBRIUM  OF  PHASES.  211 

where 

M:  +  M,'  +  ...  +  Mn  =  M'. 

There  are  as  many  relations  in  the  form  (5)  as  there 
are  phases  which  exist  beside  one  another,  —  let  us  say  r. 
The  elimination  of  F  gives  us  therefore  r  —  i  relations 
between  the  differentials  dO,  dP,  dU^  dII9,.  .  .  dllnj 
which  must  be  fulfilled  if  these  changes  are  not  to  dis- 
turb the  equilibrium  between  the  phases. 

Together  with  the  constants  of  the  total  mass, 
there  are  also  r  conditions  which  determine  the  mass  of 
each  of  the  r  phases,  as  in  the  previous  chapter  we 
found  to  be  a  result  of  experiment. 

If  only  one  substance  is  present  in  two  phases,  then 

dF=  -  SdO  +  V'dP-M'dll 
=  _  S"dB  +  V"dP  -  M"dn 
M'  =  M" 
M,f  =  M,"; 
therefore 

(6)  (S"  -  S')dB  =  (V"  -  V')dP, 

an  equation  that  leads  to  relation  (7),  page  81. 

For  two  substances  in  two  phases  we  have  the 
equations 

dF  =  -  S'de  -\-vdP- 
=  _  s'de  +  VdP  - 


and  they  give 

(7)  (S"-  sf)d8  =(v-  vf)dP-(M^'-M^d(nl  -  nt). 

If  M,"  =  M^  then  M,"  =  Mj\  i.e.,  if  both   phases 
have  the  same  ratio  of  mixture,  then  the  equation  is  the 


2  12   PRINCIPLES   OF   MATHEMATICAL    CHEMISTRY. 

same  as  for  a  single  substance.  If  this  ratio  is  differ- 
ent, then  beside  the  given  four  possible  changes,  dO, 
dP,  dLTiy  dll^,  the  relation  exists  that  the  amount  of 
the  mixture  open  to  change  remains  constant,  as  for 
example  i  gram.  It  will  therefore  be  possible  to  make 
all  dependent  upon  the  two  variables  6  and  P.  The 
above  equation  can  therefore  be  written 


=  [V"  -  V  -  (M,"- 


If,  by  a  certain  ratio  of  mixture,  the  factor  of  dP 

d6 
on  the  right  becomes  o,  therefore  -rp  =  o,  the  curve 

of  vapor  tension  has  a  tangent  parallel  to  the  axis  of 
pressure,  a  case  to  which  Duhem  refers  certain  phe- 
nomena that  are  shown  by  the  liquefaction  of  mixtures 
of  gases. 

Instead  of  pressure  and  temperature,  we  can  also 
use  pressure  and  ratio  of  mixture  as  independent 
variables,  where  the  latter  is  given  by  J//.  We  obtain 


(S"  -  SJP+dM*  =  (V"  -  V'}dP 


30 
At     constant     pressure     dP  =  o,     ~rjuTi  —   °>     ^ 

M"  =  ^//;  i.e.,  the  temperature  at  constant  pressure 


THE  EQUILIBRIUM   OF  PHASES.  213 

is   a   maximum   or  minimum   for  the  case  that   both 
phases  show  the  same  ratio  of  mixture  (Gibbs). 

In  a  corresponding  manner  we  can  prove  by  intro- 
duction of  6  and  M/  as  independent  variables  that  by 
all  mixtures,  at  the  same  temperature,  the  one  in  which 
both  phases  have  the  same  ratio,  shows  the  greatest  or 
the  smallest  pressure. 


CHAPTER    III. 

CHEMICAL   REACTIONS  THAT  DEPEND   UPON   SEVERAL 
PARAMETERS. 

THE  basis  of  all  calculations  in  chemistry  is  the 
measuring  of  the  amounts  of  the  reacting  substance 
according  to  the  different  units ;  each  chemically  dif- 
ferent substance  has  a  different  amount  for  its  mass 
which  is  called  a  mol,  and  is  used  together  with  its  frac- 
tions, the  atomic  and  equivalent  weights.  The  mass 
occurs  in  all  our  energetic  equations  in  the  form  that 
its  differential  is  multiplied  into  the  chemical  intensity; 
that  is,  as  a  quantity  or  capacity  function  (see  page  43) 
of  chemical  energy. 

Chemistry,  from  a  mathematical  standpoint,  has  up 
to  the  last  ten  years  done  nothing  but  work  out 
the  measurement  of  amounts  of  substance.  By  the 
development  ^of  the  idea  of  molecular  weight  it  has 
succeeded  in  representing  the  majority  of  chemical 
processes  as  a  function  of  one  independent  parameter, 
i.e.,  it  has  found  how  to  measure  the  amounts,  so  that 
when  we  know  the  change  that  an  amount  of  substance 
undergoes  by  a  process  we  can  give  that  for  all  the  other 
reacting  substances.  We  have  united  in  this  way,  in 
our  equations  of  simple  chemical  reactions,  the  terms  of 
chemical  energy  to  a  single  term  of  the  form  TI0dM^ 

214 


SEVERAL   PARAMETERS  IN  REACTIONS.       11 5 

As  the  later  mathematical  development  of  chemistry 
has  turned  all  its  attention  to  the  intensity  factor  17 
and  taken  for  granted  that  the  quantity  factor  M  is 
well  known,  so  have  we,  in  our  explanations,  busied 
ourselves  but  little  with  stoichiometry.  For  example, 
we  have  had  no  chance  to  study  the  hypotheses  of  the 
molecular  constitution  of  matter,  which  has  only  re- 
tained its  place  for  the  ease  by  which  it  allows  us  to 
review  the  relations  of  mass. 

It  will  be  well,  therefore,  in  concluding,  to  investi- 
gate stoichiometry,  that  oldest  branch  of  mathematical 
chemistry  which  concerns  those  processes  which  do  not 
depend  upon  a  single  parameter. 

The  reactions 

H  +  Cl  =  HC1, 
2H  +  0  =  HaO 

are  two  processes,  each  of  which  depends  upon  a  single 
parameter. 

If  we  measure  each  of  the  occurring  substances  in 
mols,  and  if  in  a  very  small  period  of  time  H  reacts 
with  x,  Cl  with  y,  and  O  with  z  atoms,  HC1  with  u  and 
H2O  with  v  mols,  then  it  is  necessary  that 

(i)  x=y=u,     -  =  z  =  v; 

hence  the  proportions  of  the  reacting  substances  are 
determined,  and  only  one  of  the  number  is  necessary  to 
find  the  others. 

If,  however,  we  have  a  mixture  of  H,  Cl,  and  O, 
there  remains,  even  when  we  are  certain  that  only  the 
above  reactions  take  place,  an  uncertainty.  The  amount 


2l6  PRINCIPLES  OF  MATHEMATICAL   CHEMISTRY. 

of  hydrogen  falls  now  into  two  parts,  in  x^  and  x^  mols, 
so  that 

(2)  x  =  x^  +  *„     %i  =  y  =  u,     x^  —  2z  —  2v 

when  the  reaction  goes  smoothly,  i.e.,  when  no  Cl,  H, 
or  O  remains  over.  It  also  follows  from 

(3)  x .  H  +y  .  Cl  +  z  .  O  =  fcHCl  +  v .  HaO, 
by  the  law  of  the  conservation  of  matter,  that 

(4)  #,  =  u  +  2v,    y  =  u,     z  —  v, 

as  before.  Here,  therefore,  the  ratio  is  not  completely 
determined,  two  of  the  quantities,  say  x  and  y,  can  be 
arbitrarily  chosen  ;  the  chemical  reaction  is  here  a 
process  depending  upon  two  parameters. 

Of  course  the  choice  is  limited  in  that  all  the  quan- 
tities must  be  positive.     If  we  choose  x  =  I,  then  for 
y  and  z  (and  also  for  u  and  v)  the  condition  must  be 
z  fulfilled  that  y  -j-  2z  =  I  for  posi- 

tive y  and  z.     If  we  assume  y  and 
z  as  coordinates  (Fig.  15),  then  the 
Y  condition  y  -\-  2z  =  I  is  fulfilled  by 
A\*all  points  on  a. straight  line,  and 
FIG.  15.  the  further  condition  of  positive 

value,  by  the  points  of  this  line  which  are  in  the  first 
quadrant  between  the  points  AB  of  the  line.  We  can, 
after  choosing  x  =  I,  choose  y  only  smaller  than  I  and 
z  smaller  than  1/2. 

According  to  this  it  is  apparent  that  the  reacting 
amounts  of  substance  in  each  chemical  reaction  is 
stoichiometrically  given  by  equations  of  the  form  (4), 
of  which  there  are  as  many  as  of  chemical  elements  in 


SEVERAL  PARAMETERS  IN  REACTIONS.       21? 

the  reaction.  If  this  number  is  n  we  have  a  simple 
chemical  reaction,  when  n  —  I  substances  enter  into 
the  reaction  equation  (3).  If  s  substances  enter,  then 
the  equation  depends  upon  s  —  n  parameters,  for  the 
n  equations  of  each  kind  contain  s  —  I  proportions,  and 
allow  n  of  them  to  be  made  dependent  upon  the  other 
s  _  i  _  n  proportions ;  hence  upon  s  —  n  arbitrary 
constants. 

In  the  same  way  (as  above  example)  that  all  possi- 
bilities of  a  reaction,  depending  upon  two  parameters, 
can  be  shown  by  the  points  of  a  line  in  the  first 
quadrant,  so  can  the  points  of  a  plane  lying  in  the  first 
octant  show  the  possibilities  of  reactions  depending 
upon  three  parameters. 

From  the  reaction 

^H  +^C1  -+  zQ  +  /N  =  «HC1  +  ^HaO  +  ze/NH, 

follows  the  condition 

x  —  u  +  2v  -f-  3«/,    y  =  u,    t—w,    z  —  v, 

that  is,  four  conditions  for  seven  unknown  quantities. 
If  we  choose  x  =  I  and  u  and  v  arbitrarily,  then 

since 

u  -|-  2v  +  3w  =  I 

w  must  be   a  positive  number.     If  we  imagine  the  w 

axis  perpendicular  to  the  plane  of  the 

paper  («,  v\  then  only  those  values  of 

u  and  v  are  available  which  give   a 

positive  value  for  w,  that  is,  give  a 

point  in   the   surface    lying    in    the  FIG.  16. 

positive  octant ;  these  points  are  those  in  the  shaded 

triangle  oA  B  (Fig.  16). 


218  PRINCIPLES   OF  MATHEMATICAL    CHEMISTRY. 

We  see  from  this  that  for  three  independent  vari- 
ables we  must  give  up  these  considerations. 

It  is  enough  to  observe  that 

u  -f-  2v  —  I  =  —  ^w 

must  be  a  negative  quantity,  and  that  is  the  case  for  all 
points  u,  v,  of  the  triangle  oAB,  while  for  the  points  in 
the  line  AB  the  trinomial  u  -f-  2v  —  I  =  o,  is  positive. 

For  four  variables  it  is  in  this  way  possible  to 
obtain  help.  It  will  be  best  illustrated  by  the  exam- 
ple to  which  this  method  was  first  applied.  Debus  * 
(1882-91)  treated  the  reaction  of  gunpowder  as  a  chemi- 
cal reaction  which  depends  upon  several  variables,  and 
Nickel  f  treated  other  chemical  technical  processes  from 
the  same  standpoint. 

Potassium  nitrate,  carbon  (charcoal),  and  sulphur 
form  by  explosion  potassium  carbonate,  sulphate,  per- 
sulphid,  carbonic  acid  gas,  carbon  monoxide  and 
nitrogen,  according  to  the  reaction 

3  +  jC  +  *S  =  /K2C03  +  «K,S04  +  ^K2S2 


The  equation  is  so  written  that  one  element,  N, 
and  one  of  the  unknown  quantities,  viz.,  the  number  of 
mols  of  N,  drop  out  of  the  equation  ;  so  that  for  the 
eight  unknowns  and  the  four  elements  we  have  the  four 
conditions 

X  =  2t  -f-  211  +  2V, 

y  —    t+w  +  a, 

Z  =    U  -\-  2Vy 

^x  =  $t  +  4U  -\-2w-\-a\ 

*Ann.  d.  chem.  212-265. 

f  Zeit.  f.  phys.  chem,  1892  ff. 


SEVERAL  PARAMETERS  IN  REACTIONS.       219 

from  which  follows  that 

;/  —         x  +  2y  —  4z  —  a, 


14^  =  —  5*  +  47  +  62  —  2a, 
jiu  =    —  x  +  57  +  42  —  6a. 

We  choose,  with  Debus,  x  —  16,  and  ask  how  the 
other  substances  are  to  be  chosen  so  that  the  reaction 
may  be  complete.  We  make  use  of  the  coordinate 
system  (Fig.  17)  y  and  z  in  the 
plane  of  the  paper  and  a  perpen- 
dicular  to  it.  In  this  the  planes 


-f-  2y  —  42  —  a, 
—  47  +   2  +  2# 


o  =  1  6 
o  =  80 
o  =  — 

are   to  be  marked,  whose   (Tafel- 

spuren)  intersection  forms  the  tri- 

angle  PQR.       The   planes   them- 

selves form  a  prism,  whose  edges 

project  in  the  direction  of  y,  and 

only  those  points  lying  within  the 

prism    have  the   coordinates  yt  z, 

a,  which  make  the  right  sides  of 

the  above  equations  positive,  and 

therefore  /,  u  and  v.     In    order   that  w  may  also  be 

positive,  the  points  must  be  chosen  below  the  surface 

o  =  —  16  -f-  57  -|-  4,3-  —  6a 

which  belong  to  the  points  of  the  prism  that  are  pro- 
jected  toward   P'Q'JK'  and  which   is   formed    by    the 


22O  PRINCIPLES  OF   MATHEMATICAL   CHEMISTRY. 

intersection  of  the  line  w  =  o, — as  is  shown  in  the 
first  principles  of  the  solid  analytical  geometry. 

We  can  now  give  by  aid  of  the  figure  the  amounts 
of  substances  necessary  for  a  complete  reaction.  It  is, 
for  16  mols  saltpeter,  as  many  mols  of  C  and  5  as  the 
coordinates  of  the  points  show  which  lie  within  the 
pentagon  P.  QRR'P'. 

Debus  limited  himself  to  the  amounts  which  were 
necessary  for  complete  reaction  without  formation  of 
carbon  monoxide,  by  which  a  =  o.  It  is  shown  by  the 
triangle  PQR\  and  Debus  has  proved  that  all  the  new 
powders  belong  to  this  plane,  while  the  older  mixtures 
are  represented  by  points  outside  of  it. 

The  volume  of  gas  developed  (page  27)  is 

V  —  \w-\-a-\-  -J224OO  =  — - — (iox-{-2oy-\-i62-\-4a)c<:. 

\  2,'  2o 

for  one  of  the  composition 

G  —  x .  101  -\-  y .  12  -\-  z .  32  grams  ; 

and  the  same  quantity  gave  an  amount  of  heat  equal 
to 

W  =  t .  279530  +  2/344640  +  2/108000 
+  ^97000  +  #  29000  —  x\  19480 
=  144196^  —  169297  —  8783,2  —  11036^, 

where  the  heats  of  formation  of  the  reacting  substances 
occur  as  functions  of  the  number  of  mols. 

If  we  place  again  x  —  16  and  a  =  o,  it  follows,  as 
is  not  necessary  to  develop  here,  that  the  greatest 


SEVERAL   PARAMETERS  IN  REACTIONS.       221 

volume  development  takes  place  in  point  Pt  the  greatest 
development  of  heat  in  point  Q,  as  well  when  we  use 
powders  the  same  weights  and  also  when  they  contain 
the  same  amounts  of  saltpeter.  In  the  first  case  we 
find  the  greatest  value  for  A  =  v:g  and  A'  =  w\g\  in 
the  second  the  greatest  value  for  Fand  W« 

Here  A  is  proportional  to  the  ratio  in  which  the 
angle,  formed  by  the  lines  V  •=•  o  and  g  =  o,  is  divided 
by  the  line  V  —  \g  =  o,  and  A/  has  corresponding 
values. 


INDEX. 


Absolute  mass 21,  78,  98 

Absolute  temperature 26 

Absorption 128 

Activity 184 

Adiabatic 51,  53 

Aggregation,  Changes  of 82 

Allotropic  change 88 

Ammonium  carbimate 158 

Ammonium  chloride 89 

Ampere 96 

A  vogadro • 26,  68 

Arrhenius 123,  127,  155,  169,  186 

Beckmann 168 

Bertholot .* II,  74 

Boiling-point,  Raising  of  the - 161,  168 

Calcite 9° 

Calorie 4,  5,  6 

Capacity 43 

Carbon 13 

Carbon  dioxide 147 

Carbon  disulphide 168 

Carnot 47,  5<3 

Catalysis 197 

Chemometer, , , , 106 

223 


224  INDEX. 

Clapeyron 82 

Clark  element 104 

Clausius 65 

Combustion,  Heat  of 35 

Concentration 69 

Concentration  product 139,  194 

Conductivity,  Electrical ....-..- 109,  156,  189 

Constituent 127 

Coulomb 101 

Cycle 56 

Daniell  element 103 

Debus 218,  219,  220 

Deville 89,  1 5 1 

Dieterici 5,  23,  32,  86 

Diffusion 182 

Dilution,  Heat  of 94 

Dissociation 89,  123,  156,  169 

Dissociation,  Degree  of 109,  148,  157,  169 

Dissociation,  Electrolytic 156,  169 

Duhem 74,  212 

Dyne 21 

Electromotive  force .97,  127 

Elements,  Galvanic 102 

Enantiotropic 88 

Energy,  Electrical 96 

Energy,  Factors  of 41 

Energy,  Free .31,  74,  210 

Energy,  Intrinsic 2 

Energy,  Kinetic 19,  182 

Energy  of  motion 19 

Energy,  Potential 20 

Energy,  Conservation  of i,  2 

Entropy  41,  70,  116 

Equilibrium,  Chemical 141,  196 

Equilibrium,  Complete 206 

Equilibrium  of  phases  .,.,..., » 


INDEX.  22$ 

Equivalent  ot  heat 23 

Equivalents IOI>  I42 

ErS 21,23 

Ester Ig8i  200 

Exchange  numbers I3I 

Faraday IOO 

Fick !  8  3 

Force,  Unit  of 21 

Free  energy 31,  74,  210 

Freedom,  Degrees  of 202 

Freezing-point,  Lowering  of 161,  169 

Gas  constant 25,  48,  67 

Gas  mixtures 67 

Gases 25 

Gibbs 70,  75,  78,  102,  108,  127,  150,  205,  209,  213 

Gockel 103 

Graham 175 

Guldberg 143,  196 

Gunpowder 218 

Heat  of  formation 12,  35,  36 

v.  Hebnkoltz 102 

Henry 129 

Hess 3 

Highly  diluted 162 

Hittorf* : ill 

varit  Hoff 88,  138,  166,  178 

Horstmann 10,  74,  91 

Hydrochloric  acid  1 1 

Hydriodic  acid 143 

Intensity 41 

Intensity,  Chemical 78,  106,  114,  121,  181 

Inversion 196 

Ions 109,  1 86 

Isambcrt 159 


226  INDEX. 

Isolated  system 76,  1 16 

Isothermal 51,  52,  53 

Jahn !  o 

Joule 24,  98 

Kinetic  energy .19,  182 

Kirchhoff , 18,  94 

Kohlrausch 108,  157,  189 

Laplace > 3 

Lavoisier 2 

Le  Chatelier 90,  153,  160 

Lehmann 88 

Lemoine 145 

Loeb 113 

Mariotte-Gay-Lussac  Law,  The 48 

Mol 6 

Molecular  volume 26 

Monotropic 88 

Naumann 33 

Nernst 113,  145,  199 

Neumann .V 5  7,  70 

Nickel 218 

Nitrogen  tetroxide 150 

Ohm 96 

Osmotic  pressure 174,  181 

Ostvvald 5,  6,  21,  41,  106,  197 

Overcooling 87 

Parameters ; 2,  132,  215,  216 

Partial  pressure 68 

Partition,  Semi-permeable 175 

P<?an  de  St.  Gilles 160 

Pfeffer 175,178 

Phases 75 


INDEX.  227 

Phases,  Rule  of 202 

Phosphorus , 168 

Planck u,  69 

Poincare. 66 

Potential,  Electrical 97,  98 

Potential,  Thermodynamical 74,  121,  210 

Potential  energy 20 

Pressure,  Osmotic 174,  181 

Pressure 22 

Raoult 166 

Reaction  equivalent 142 

Reaction,  Heat  of 8,  31,  145 

Reaction,  Simple 131 

Regnault 16,  32,  84,  95 

Reicher 88 

Resistance,  Electrical 97,  98 

Reversible eg 

Roozeooom „  „ .  206,  207 

Salt,  Common , 112,  169 

Saponification , 198 

Scheffer 191 

Semi-permeable  partition 175 

Specific  gravity 26 

Specific  volume 26 

State,  Equation  of, 25,  49 

Steam-engine 57 

Stohmann r 40 

Stoichiometry , 6,  215 

Sugar , . , 178,  196 

Sulphuric  acid 9 

Sulphurous  acid 206,  207 

Thermo-neutrality 139 

Thomsen 9,  40,  83,  95 

Thomson 83 

Transference  numbers in 


228  INDEX. 

Troost 151 

Turpentine,  Oil  of 167 

Vapor  tension,  Lowering  of 161,  164,  174 

Vapor  tension,  Curve  of 86 

Velocity  of  change 192 

Visser 83 

Volt 97 

Volt-ampere 98 

Volume  energy 21,  22,  25 

Waage I43>  196 

Wandering  velocity no 

Water 16,  32,  39,  82,  94,  206,  207 

Watt 98 

Wiedemann,  G. 90 

Wilhelmy 196 

Zeuner 16,  84,  95 

Zinc  sulfate qo 


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8 


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9 


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11 


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12 


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Williams's  Lithology ' Svo,  3  00 

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13 


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"           Philosophy  of  the  Steam  Engine 12mo,  75 

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Construction,  and  Operation 8vo,  7  50 

2  parts,  12  00 

Rontgen's  Thermodynamics.     (Du  Bois. ) 8vo,  5  00 

Peabody's  Thermodynamics  of  the  Steam  Engine 8vo,  5  00 

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Wood's  Thermodynamics,  Heat  Motors,  etc 8vo,  4  00 

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Trow  bridge's  Stationary  Steam  Engines , .  .4to,  boards,  2  50 

14 


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Fisher's  Table  of  Cubic  Yards. c Cardboard,  25 

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Beard's  Ventilation  of  Mines 12mo,  2  50 

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niSCELLANEOUS  PUBLICATIONS, 

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J5 


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